# Asymptotics of the partition function for $\beta$-ensembles at high temperature

Charlie Dworaczek Guera

► To cite this version:

Charlie Dworaczek Guera. Asymptotics of the partition function for $\beta$-ensembles at high temperature.
2024. hal-04674202

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https://hal.science/hal-04674202v1

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# Asymptotics of the partition function for $\beta$-ensembles at high temperature *

Charlie Dworaczek Guera†

## Abstract

We consider a model for a gas of $N$ confined particles interacting via a two-body logarithmic interaction, namely the real $\beta$-ensembles. We are interested in the regime where the inverse temperature scales as $N\beta = 2P$ with $P$ a fixed positive parameter; this is called the high-temperature regime. The confining potential is of the form $x^2 + \phi$ with bounded smooth function $\phi$. We establish for this model, the existence of a large-$N$ asymptotic expansion for the associated partition function. We also prove the existence of a large-$N$ asymptotic expansion of linear statistics for general confining potentials. Our method is based on the analysis of the loop equations. Finally, we establish a continuity result for the equilibrium density with respect to the potential dependence.

## Contents

<table><tr><td><b>1 Introduction</b></td><td><b>2</b></td></tr><tr><td>1.1 Setting of the problem</td><td>2</td></tr><tr><td>1.2 Assumptions</td><td>5</td></tr><tr><td>1.3 Main results</td><td>6</td></tr><tr><td>1.4 Outline of the proof</td><td>8</td></tr><tr><td>1.5 Notations and conventions</td><td>9</td></tr><tr><td><b>2 A priori bound on the linear statistics</b></td><td><b>11</b></td></tr><tr><td>2.1 Equilibrium measure</td><td>11</td></tr><tr><td>2.2 Concentration inequality</td><td>11</td></tr><tr><td>2.3 A priori bound on linear statistics</td><td>12</td></tr><tr><td><b>3 Properties and control of the operators involved</b></td><td><b>14</b></td></tr><tr><td>3.1 Definitions</td><td>14</td></tr><tr><td>3.2 Control on the non-commutative derivative operator</td><td>14</td></tr><tr><td><b>4 Control on the master operator Ξ</b></td><td><b>16</b></td></tr><tr><td>4.1 Definition</td><td>17</td></tr><tr><td>4.2 Preliminaries</td><td>17</td></tr><tr><td>4.3 Closed form for Ξ<sup>-1</sup></td><td>19</td></tr><tr><td>4.4 Controls on the inverse of the master operator</td><td>22</td></tr></table>

<footnote>*This project has received funding from the European Research Council (ERC) under the European Union Horizon 2020 research and innovation program (grant agreement No. 884584).</footnote>

<footnote>†ENSL, CNRS, France</footnote>

<footnote>email: charlie.dworaczek@ens-lyon.fr</footnote>
<table>
  <tbody>
    <tr>
      <td>5</td>
      <td>Asymptotic expansion of the linear statistics</td>
      <td>36</td>
    </tr>
    <tr>
      <td>5.1</td>
      <td>Loop equations for general functions</td>
      <td>36</td>
    </tr>
    <tr>
      <td>5.2</td>
      <td>Asymptotic expansion of linear statistics</td>
      <td>36</td>
    </tr>
    <tr>
      <td>6</td>
      <td>Parameter continuity of the equilibrium measure</td>
      <td>39</td>
    </tr>
    <tr>
      <td>6.1</td>
      <td>Setting for Banach fixed-point theorem</td>
      <td>40</td>
    </tr>
    <tr>
      <td>6.2</td>
      <td>Regularity of the equilibrium measure</td>
      <td>47</td>
    </tr>
    <tr>
      <td>7</td>
      <td>Proof of Theorem 1.4</td>
      <td>48</td>
    </tr>
    <tr>
      <td>7.1</td>
      <td>Asymptotic expansion of the partition function for the Gaussian potential</td>
      <td>48</td>
    </tr>
    <tr>
      <td>7.2</td>
      <td>Free energy of the model</td>
      <td>51</td>
    </tr>
    <tr>
      <td>7.3</td>
      <td>Interpolation with general potential</td>
      <td>52</td>
    </tr>
    <tr>
      <td>8</td>
      <td>Conclusion</td>
      <td>53</td>
    </tr>
    <tr>
      <td>A</td>
      <td>Appendix: Lemmas and technical results</td>
      <td>53</td>
    </tr>
    <tr>
      <td>B</td>
      <td>Integrability of the constants</td>
      <td>56</td>
    </tr>
    <tr>
      <td>B.1</td>
      <td>Parameter continuity of norms of certain functions</td>
      <td>56</td>
    </tr>
    <tr>
      <td>B.2</td>
      <td>Parameter-continuity of C<sub>L,t</sub> and K<sub>V<sub>G,φ,t</sub></sub></td>
      <td>62</td>
    </tr>
    <tr>
      <td>References</td>
      <td></td>
      <td>63</td>
    </tr>
  </tbody>
</table>

# 1 Introduction

## 1.1 Setting of the problem

Let $P > 0$ and $V$ be a function growing sufficiently fast at infinity. The real $\beta$-ensemble at high temperature is the particle system on $\mathbb{R}$, $\{x_i\}_{i=1}^N$ with the following distribution:

$$d\mathbb{P}_N^V(\underline{x}) = p_N^V(\underline{x})dx_1 \dots dx_N \quad \text{with} \quad p_N^V(\underline{x}) := \frac{1}{\mathcal{Z}_N[V]} \prod_{i<j}^N |x_i - x_j|^{2P/N} \prod_{i=1}^N e^{-V(x_i)} \quad (1)$$

where $\mathcal{Z}_N[V] > 0$ is the partition function that ensures that $\mathbb{P}_N^V$ is a probability measure on $\mathbb{R}^N$, namely

$$\mathcal{Z}_N[V] = \int_{\mathbb{R}^N} \prod_{i<j}^N |x_i - x_j|^{2P/N} \prod_{i=1}^N e^{-V(x_i)} dx_i. \quad (2)$$

Here, the factor 2 in the two-body interaction is irrelevant and just makes the equations "nicer". The main goal of this article is to establish the existence of the large $N$-asymptotic expansion of $\log \mathcal{Z}_N[V]$ under some assumptions on $V$ using the technique first used in [ACM92] and [ACKM93] and later developed in [BG13a], [BG13b]. Note that when $2P/N$ is replaced by a $N$-independent $\beta \ge 0$ (that one can interpret as a coupling constant that measures the strength of the interaction), the distribution is known as the real $\beta$-ensemble in the fixed temperature regime and represents for polynomial potential the joint law of eigenvalues of the so-called Orthogonal (resp. Unitary, resp. Symplectic) ensemble for $\beta = 1$ (resp. $\beta = 2$, resp. $\beta = 4$) (see [AGZ10]). For general $\beta \ge 0$ and quadratic $V$, the $\beta$-ensemble was expressed as the law of the spectrum of tridiagonal random matrices with independent entries [DE02]. The result was then extended to general polynomial $V$ in [KRV16].
When $\beta$ is fixed, a great deal is known about this model because of twenty years of intensive study. First, central limit theorems were proven in [Joh98], [BG13a] [Shc14], [BLS18], [Lam21a], [LLW19], asymptotic of the partition function in [BG13a], [BG13b], [BGK15], local laws [BYY14], [BEY14], [CFLW21], [BMP22] [Pei22] and universality results [PS97], [DKM+99], [DG07a], [DG07b]. For $\beta = 2$ constant and $V$ polynomial, the asymptotic expansion of the partition function has the form

$$N^{-2} \log Z_N^{(2)}[V] \sim \sum_{g \ge 0} N^{-2g} c_g$$

where the previous equality has to be understood in the sense of an asymptotic expansion. The coefficients $(c_g)_{g \ge 0}$ of this expansion correspond to enumerations of maps and, more generally, the asymptotic expansion of $\log Z_N^{\beta}[V]$ gives information on the enumerations of graphs embedded in surfaces [Mar14], [MTY05]. In [DFGZ23], the authors were able to establish the asymptotics for the moments of Hilbert-Schmidt norms of matrices uniformly distributed on unit balls. They were able to link these moments with the partition function of $\beta$-ensembles with singular potential, for which it is possible to obtain the leading asymptotics.

The study of the $\beta$-ensembles at high temperature has attracted a lot of attention recently since links were discovered with integrable systems, including the famous classical Toda chain [Tod67]. The integrable structure of this system, namely the existence of a sufficient number of conserved quantities, can be established by the existence of a so-called Lax matrix, whose spectrum is invariant under the dynamics. At long times, the model doesn't thermalize, ie it doesn't reach thermal equilibrium but is rather described by a more sophisticated probability measure called the Generalized Gibbs Ensemble (GGE) [Jay57]. This is due to the existence of a set of locally conserved quantities, which highly constrains the dynamics. In the context of the Toda chain, the GGE has been studied in [Spo20] and a link was established with the Gaussian $\beta$-ensembles. In the case of a Gaussian potential, it was shown that the distribution of the Lax matrix under the GGE was similar to the law of the tridiagonal representation of the Gaussian $\beta$-ensembles of Dumitriu and Edelman. This link was explored in [GM22] for more general potentials via large deviation techniques.

However, the high temperature regime had already been the subject of research with the pioneering works [CL97], [BG99] and [ABG12]. More recently, large deviation principles (LDP) [Pak20], [GZ19] and central limit theorems for $\beta$-ensembles at high temperature were shown in the circular case [HL21]. In the real case [NT18], such a result was obtained for quadratic potentials and polynomial test-functions, for general potentials with smooth bounded functions in [DGM23] and for polynomial potentials and polynomial test-functions in [MM23]. Convergence results of bulk local statistics to Poisson point-processes were established, first in [BGP15] and later in [NT18], for Gaussian potential, then generalized to general potentials in [NT20] and finally to general interactions, potential and geometric setting in [Lam21b]. The latter work, includes the convergence of the edge statistics and the asymptotic law of the edge of the spectrum, namely a Gumbel distribution which was first discovered in the quadratic case by [Pak18]. This is in adequation with the fact that this law can be seen as the limit of the $\beta$-Tracy Widom law when $\beta \to 0$ [AD14]. Finally, in [FM21], the authors analyzed the so-called loop equations to deduce the moments of the subdominant correction of the equilibrium measures corresponding to Gaussian, Laguerre and Jacobi ensembles. A natural extension of all these results is to obtain the large $N$ asymptotic expansion of the $n$-linear statistics $\langle f \rangle_n^{\beta}$ for general test-functions $f$ where $L_N := N^{-1} \sum_{a=1}^N \delta_{x_a}$ is the empirical distribution, $\mathcal{L}_N := L_N - \mu_V$ and

$$\langle f \rangle_{\mu_1 \otimes \dots \otimes \mu_k}^V := \mathbb{E}_N^V \left[ \int_{\mathbb{R}} f(x_1, \dots, x_k) \prod_{i=1}^k d\mu_i(x_i) \right]. \qquad (3)$$
By [GZ19], the sequence of random probability measures $(L_N)_N$ satisfies a large deviation principle at speed $N$ with a strictly convex, good rate function $I$. The latter is defined by $I := \mathcal{E} - \inf_{\mu \in \mathcal{M}_1(\mathbb{R})} \mathcal{E}(\mu)$ where, for all absolutely continuous probability measures $\mu$ with respect to the Lebesgue measure:

$$ \mathcal{E}(\mu) := \int_{\mathbb{R}} V d\mu - 2P \iint_{\mathbb{R}^2} \log |x - y| d\mu(x)d\mu(y) + \int_{\mathbb{R}} \log \left( \frac{d\mu}{dx} \right) d\mu $$

and $+\infty$ for other probability measures on $\mathbb{R}$. In this functional, the last term is called the entropy of the measure $\mu$, and because of conventions, it represents the negative physical entropy. This term is negligible when the inverse temperature $\beta$ is constant and the potential is scaled by $N$ (it doesn't appear in the corresponding good rate function [AGZ10]) but has the same order of magnitude as the energy when the correlation between particles is small/the temperature is high (see [Dea08][34] for a discussion). Minimizing the functional $\mathcal{E}$ amounts to minimizing the energy while maximizing the (physical) entropy.

We call the unique minimizer of $\mathcal{E}$ and denote it $\mu_V$ (we omit the $P$-dependence since this parameter is fixed throughout the entire article) the equilibrium measure which is Lebesgue continuous with p.d.f. $\rho_V$ characterized by

$$ V(x) - 2P \int_{\mathbb{R}} \log |x - y| \rho_V(y) dy + \log \rho_V(x) = \lambda_V \qquad x - ae \qquad (4) $$

where $\lambda_V$ is a constant (see [GM22][Lemma 3.2]). In this context, $\rho_V$ is supported on $\mathbb{R}$ because of the presence of the entropy in the minimizing equation. This is a major difference with the $\beta$-constant case and $N$-scaled potential, where the equilibrium measure is compactly supported. This equation can be rewritten, assuming $\rho_V$ is continuous (which is true as long as $V$ is smooth; see [DGM23][Lemma 2.2]), as

$$ \forall x \in \mathbb{R}, \rho_V(x) = \exp \left( -V(x) - 2P U^{\rho_V}(x) + \lambda_V \right), \qquad U^{\rho_V}(x) := - \int_{\mathbb{R}} \log |x - y| \rho_V(y) dy. \quad (5) $$

One observes that $U^{\rho_V}$ diverges logarithmically to $-\infty$ at infinity [DGM23][Lemma 2.4]. Hence, assuming that $V$ grows fast enough at infinity, instead of a compactly supported measure as in the constant $\beta$ case, the equilibrium density is an exponentially fast decaying function at infinity in the case $\beta N = 2P$. Furthermore, this measure can be seen as an interpolation (with the appropriate scaling in $P$) called originally the Gauss-Wigner crossover [ABG12] between the equilibrium of the classical $\beta$-ensembles (when $P$ goes to $+\infty$) and the measure $d\mu(x) = e^{-V(x)}dx/Z$ (when $P$ goes to 0) [NT20]. Note that under the choice of $V_G(x) = x^2/2$, the density has an explicit form [ABG12]:

$$ \rho_{V_G}(x) = \frac{e^{-\frac{x^2}{2}}}{\sqrt{2\pi} |\hat{f}_{\alpha}(x)|^2}, \qquad \hat{f}_{\alpha}(x) := \sqrt{\frac{P}{\Gamma(P)}} \int_{0}^{+\infty} t^{P-1} e^{-\frac{t^2}{2}+ixt} dt. $$

This density is also explicit in some other models, namely the Laguerre and Jacobi ensembles, involving hypergeometric functions [Maz22].

Our goal here is to follow a strategy introduced in [BG13a], [BG13b] and used in [BGK15] and [BGK16] to establish the existence of the large $N$-asymptotic of $\log \mathcal{Z}_N[V]$ for a general potential of the form $V_{G,\phi}(x) := x^2/2 + \phi$ where $\phi$ is a bounded smooth function. Namely we wish to show that for all $K \ge 0$, there exists $c_0, \dots, c_K \in \mathbb{R}$ depending on $\phi$ and $P$ such that

$$ \frac{1}{N} \log \mathcal{Z}_N[V_{G,\phi}] = \sum_{i=0}^{K} \frac{c_i}{N^i} + O\left(N^{-(K+1)}\right). $$
Before stating the main results and explaining how to obtain them, we need to introduce some objects. An object that appears naturally when tackling this model is the so-called master operator Ξ defined, for sufficiently smooth $\phi$, by:

$$ \forall x \in \mathbb{R}, \quad \Xi[\phi](x) := \phi'(x) + (\log \rho_V)'(x)\phi(x) + 2P \left( \mathcal{H}[\phi\rho_V](x) - \int_{\mathbb{R}} \mathcal{H}[\phi\rho_V](y)d\mu_V(y) \right) \quad (6) $$

where $\mathcal{H}$ denotes the Hilbert transform, which is defined by $\mathcal{H}[f](x) = \int_{\mathbb{R}} \frac{f(y)}{y-x}dy$ and where the integral has to be understood as a Cauchy principal value. The main difference between $\Xi$ and the master operator $\mathcal{K}$ that arises in the classical $\beta$-ensembles is the first derivative term of the RHS in (6). Because of this term, $\Xi$ is then an unbounded operator. While in the classical $\beta$-ensembles, $\mathcal{K}$ is easily invertible and controled, see [BFG15] dealing such an operator demands lot of technicalities in general. In particular, inverting and obtaining controls is much more difficult in this case.

One can understand the operator $\Xi$ as controlling the fluctuations of the empirical measure $L_N$ with respect to $\mu_V$ in the sense that when one writes, for a sufficiently smooth function $\phi$,

$$ \langle \phi \rangle_{L_N}^V = \langle \phi \rangle_{\mu_V} + R_N, $$

the remainder $R_N$ is expressed as a sum of linear statistics of functions involving $\Xi^{-1}[\phi]$ and goes to zero when $N$ is large. Above, the superscript for the expectation value with respect to $\mu_V$ can be omitted since the measure is deterministic. Furthermore, as we will show in the present work, if $V_1$ and $V_2$ are two potentials, $\Xi^{-1}$ appears naturally when one wants to study the variation between the two corresponding equilibrium measures, namely $\mu_{V_1} - \mu_{V_2}$. Finally, by exploiting the fact that the partition function $Z_N[V]$, introduced in (2), is invariant under the transformation $\lambda_i \to \lambda_i + tN^{-1/2}\phi(\lambda_i)$ for $\phi$ sufficiently smooth, the authors showed a central limit theorem in [DGM23] ie that

$$ \sqrt{N} \int \Xi[\phi] d(L_N - \mu_V) \stackrel{\text{law}}{\Rightarrow} \mathcal{N}(0, \sigma^2(\phi)) $$

for $\sigma^2$ a positive quadratic form. It has been shown in this same article that this operator is invertible and that when $\phi$ is smooth, so is $\Xi^{-1}[\phi]$. This was done by inverting the operator $\mathcal{L}$, defined by $\mathcal{L}[\phi] := \Xi[\phi']$, on the Hilbert space:

$$ \mathcal{H} := \left\{ u \in L^2(\mu_V) \mid u' \in L^2(\mu_V), \int_{\mathbb{R}} u d\mu_V = 0 \right\}, \quad \langle u, v \rangle_{\mathcal{H}} := \langle u', v' \rangle_{L^2(\mu_V)}. \quad (7) $$

Once the operator $\mathcal{L}$ is inverted, this straightforwardly implies that $\Xi$ is invertible with inverse $\Xi^{-1}[\psi] = (\mathcal{L}^{-1}[\psi])'$.

The establishment of the large-$N$ behavior of the 1-linear statistics is based on the so-called loop equations or Schwinger-Dyson equations introduced in [ACM92],[ACKM93]; see [BGK16] for a more precise state of the art.

## 1.2 Assumptions

For the rest of the paper, we use the following list of assumptions on the potential $V$:

**Assumptions 1.1 The potential $V$ satisfies:**

(i) $V \in C^\infty(\mathbb{R})$,

(ii) $V(x) \underset{|x| \to +\infty}{\longrightarrow} +\infty$ and $|V'(x)| \underset{|x| \to +\infty}{\longrightarrow} +\infty$,
(iii) The measure $\mu_V$ satisfies the Poincaré inequality if there exists $C_{\text{Poinc}} > 0$ (depending on $V$ and $P$) such that for all $f \in C_c^1(\mathbb{R})$

$$ \mathrm{Var}_{\mu_V}(f) := \int_{\mathbb{R}} \left( f - \int_{\mathbb{R}} fd\mu_V \right)^2 d\mu_V \le C_{\mathrm{Poinc}} \int_{\mathbb{R}} (f')^2 d\mu_V. \quad (8) $$

(iv) For all polynomial $Q \in \mathbb{R}[X]$ and $\alpha > 0$, all $p \ge 0$, $Q(V^{(p)}(x)) e^{-V(x)} = \lim_{|x| \to \infty} o(x^{-\alpha})$.

(v) The function $\frac{1}{V'^2}$ is integrable at infinity, and $\frac{V^{(k)}(x)}{V'(x)} = O(1)$ for $k \ge 2$.

Assumption (i) is necessary to ensure that $\mu_V$ and $\Xi^{-1}[\phi]$ for $\phi$ smooth are smooth (its derivatives involve derivatives of $V$).

Assumption (ii) is sufficient condition for $\mathcal{Z}_N[V]$ to be well-defined. Indeed, the assumption on $V'$ implies that $V$ grows faster than linearly at infinity, which implies that the $\beta$-ensemble is well-defined. Another consequence is that $\rho_V$ has exponential decay at infinity. The fact that $V'$ goes to infinity is also necessary to ensure that $\Xi^{-1}[\phi]^{(k)}(x) \underset{|x| \to \infty}{\longrightarrow} 0$ for all $k \ge 0$ and for bounded smooth $\phi$.

Assumption (iii) implies that $\Xi$ defined in (6), is invertible; see [DGM23, Proposition 2.6]. The authors showed that for any potential of the form $V = V_{\text{conv}} + \phi$, where $V_{\text{conv}}$ is a strictly convex potential outside of a compact set and $\phi$ a bounded function, $\mu_V$ satisfies the Poincaré inequality.

Assumption (iv) is necessary to ensure that the equilibrium density $\rho_V$ is smooth.

Assumption (v) allows one to prove that $\Xi^{-1}$ is continuous with respect to the appropriate norms. Indeed, when differentiating $\Xi^{-1}[\phi]$, for $\phi$ a smooth function, quantities behaving at infinity like $V^{(k)}(x)V'(x)^{-1}$ will naturally arise. On the other hand, we will integrate some functions that behave like $V'(x)^{-2}$ at infinity.

These conditions are satisfied, for example, for every $V$ in the following class

$$ \left\{ x \mapsto a_n x^{2n} + \phi(x), n > 0, a_n > 0, a_i \in \mathbb{R}, \phi^{(k)} \text{ bounded } \forall k \in \mathbb{N} \right\} \cup \left\{ x \mapsto \frac{e^{\gamma x} + e^{-\gamma x}}{\alpha}, \alpha > 0, \gamma \in \mathbb{R} \right\}. $$

These potentials satisfy assumptions (iii); see [DGM23][Proposition 2.6, Remark 2.7] for a discussion about it. On the other hand, potentials like $V(x) = e^{x^2}$ violate assumption (v), therefore they do not fit in our analysis.

## 1.3 Main results

To state the next result, we recall that $\mathcal{L}_N := L_N - \mu_V$.

**Theorem 1.2 (Asymptotic expansion of linear statistics)** Under assumptions 1.1 on the potential $V$, for all smooth function $\phi \in L^2(\mathbb{R}^k)$, such that $\phi^{(j)} \in L^2(\mathbb{R}^k)$ for all $j \ge 0$, there exists a unique sequence $(b_i)_{i \ge [k/2]}$ depending on $V$, $\phi$ and $P$ such that for all $K > 0$:

$$ \langle \phi \rangle^{V} \otimes^{k} \mathcal{L}_N = \sum_{i=[k/2]}^{K} \frac{b_i}{N^i} + O(N^{-(K+1)}). $$

Our goal is to obtain the existence of an asymptotic expansion for $\mathcal{Z}_N[V_{G,\phi}]$ where $V_{G,\phi}(x) := x^2/2 + \phi(x)$ and $\phi$ is a smooth function. As will be explained further, one would like to deduce the
asymptotic expansion of $\log \mathcal{Z}_N[V_{G,\phi}]$ from an integration of the one for $t \in [0, 1] \mapsto \langle \phi \rangle_{\mathcal{L}_N}^{V_{G,\phi,t}}$ where $V_{G,\phi,t}(x) := x^2/2 + t\phi(x)$. In order to make this step rigorous, we need the following continuity result with respect to $t$.

**Theorem 1.3** Under assumptions 1.1 on the potential $V$, for all $i \in \mathbb{N}$ and for all smooth function $\phi \in L^2(\mathbb{R})$, such that $\phi^{(k)} \in L^2(\mathbb{R})$ for all $k \ge 0$,

$$ \|\rho_{V_{\phi,t}} - \rho_{V_{\phi,t'}}\|_{W_i^{\infty}(\mathbb{R})} \xrightarrow{t \to t'} 0 $$

where $V_{\phi,t}: x \mapsto V(x) + t\phi(x)$ where $t \in [0, 1]$. The $W_i^{\infty}(\mathbb{R})$-norm is defined as $\|f\|_{W_i^{\infty}(\mathbb{R})} := \max_{k \in [0,n]} \|f^{(k)}\|_{L^{\infty}(\mathbb{R})}$. Furthermore, for all $x \in \mathbb{R}$, $t \mapsto \rho_{V_{\phi,t}}(x) \in C^{\infty}(\mathbb{R})$ and satisfies the following integro-differential equation for all $t \ge 0$ and $x \in \mathbb{R}$:

$$ \partial_t \rho_{V_{\phi,t}}(x) = \left( -\phi(x) + \int_{\mathbb{R}} \phi(s) \rho_{V_{\phi,t}}(s) ds \right) \rho_{V_{\phi,t}}(x). $$

Theorem 1.2 together with Theorem 1.3 allow us to deduce the following theorem.

**Theorem 1.4 (Asymptotic expansion of the partition function)** Let $\phi \in L^2(\mathbb{R})$ be a smooth function, such that $\phi^{(k)} \in L^2(\mathbb{R})$ for all $k \ge 0$. There exists a unique sequence $(c_i)_{i \ge 0} \in \mathbb{R}^N$ depending on $\phi$ and $P$, such that for all $K \ge 0$,

$$ \frac{1}{N} \log \mathcal{Z}_N [V_{G,\phi}] = \sum_{i=0}^{K} \frac{c_i}{N^i} + O(N^{-(K+1)}). $$

The leading term $c_0$ is given by:

$$ c_0 := - \int_{\mathbb{R}} V_{G,\phi}(x) d\mu_{V_{G,\phi}}(x) + P \iint_{\mathbb{R}^2} \log|x-y| d\mu_{V_{G,\phi}}(x) d\mu_{V_{G,\phi}}(y) - \int_{\mathbb{R}} \log\left(\frac{d\mu_{V_{G,\phi}}(x)}{dx}\right) d\mu_{V_{G,\phi}}(x). \quad (9) $$

The first subleading term $c_1$ is given in terms of the Euler-Mascheroni constant $\gamma$ and $\Xi_t^{-1}$ the inverse of the master operator associated with the potential $V_{G,\phi,t}$. It can be written as

$$ c_1 := \gamma \frac{P}{2} + \frac{\log(1+P)}{2} + \frac{1}{2} \sum_{j \ge 1} \left( \log \left( 1 + \frac{P+1}{j} \right) - \log \left( 1 + \frac{1}{j} \right) - \frac{P}{j} \right) \\ - P \int_0^1 \left[ \left\langle \partial_1 \Xi_t^{-1} [\phi] \right\rangle_{\mu_{V_{G,\phi,t}}} + \left\langle \Theta^{(2)} \circ \Xi_{t,1}^{-1} \left[ \partial_2 \mathcal{D} \circ \Xi_t^{-1} \phi \right] \right\rangle_{\mu_{V_{G,\phi,t}}} \right] dt. \quad (10) $$

Above, $\Theta^{(2)}$ and $\mathcal{D}$ are explicit operators given in Section 3 while

$$ \Xi_t^{-1}[\phi](x) = \frac{1}{\rho_{V_{G,\phi,t}}(x)} \int_x^{+\infty} \mathcal{T}_t[\phi](y) d\mu_{V_{G,\phi,t}}(y) $$

where $\mathcal{T}_t$ is an explicit kernel operator given in (77). The symbol $\Xi_t^{-1}$ is also defined in Sections 3 and 4. It is a standard fact (see for example [GZ19][Theorem 1.2]) that the leading term $c_0$ is the free energy of the model ie, $c_0 = \lim_{N \to \infty} N^{-1} \log \mathcal{Z}_N[V_{G,\phi}] = -\inf_{\mu} \mathcal{E}(\mu) = -\mathcal{E}(\mu_{V_{G,\phi}})$ where the infimum runs over probability measures on $\mathbb{R}$.
## 1.4 Outline of the proof

This strategy is based on the following interpolation equation of the form:

$$ \log \mathcal{Z}_N[V_{G,\phi}] = \log \mathcal{Z}_N[V_G] - N \int_0^1 \langle \phi \rangle_{L_N}^{V_{G,\phi,t}} dt. \quad (11) $$

On the RHS, it is convenient to have $\log \mathcal{Z}_N[V_G]$ since, by Mehta's formula [Meh04, 17.6.7], one can extract its asymptotic expansion at large $N$. Once this identity is obtained, the derivation of the asymptotic expansion of the free energy of the model $\log \mathcal{Z}_N[V]$ follows from a similar expansion for the 1-linear statistics as soon as one has sufficiently precise controls on the remainder's dependence on the data of the problem.

We now explain how to derive Theorem 1.2 for a general potential $V$ satisfying assumptions 1.1. The proof is based on the analysis of the Schwinger-Dyson equations (SDE). It consists of a tower of equations that link linear statistics of different orders together. The simplest equation is the one at level 1, which reads for any $\phi$ smooth enough:

$$ \langle \phi \rangle_{\mathcal{L}_N}^V = \frac{P}{N} \left\langle \left( \Xi^{-1}[\phi] \right)' \right\rangle_{\mu_V} + \frac{P}{N} \left\langle \left( \Xi^{-1}[\phi] \right)' \right\rangle_{\mathcal{L}_N}^V - P \left\langle \mathcal{D} \circ \Xi^{-1}[\phi] \right\rangle_{\mathcal{L}_N \otimes \mathcal{L}_N}^V \quad (12) $$

where $\mathcal{D}$ is the operator defined for all $x \neq y$ by $\mathcal{D}[\phi](x, y) := \frac{\phi(x) - \phi(y)}{x - y}$. This equation links the 1-linear statistic and the 2-linear statistic $\langle \mathcal{D} \circ \Xi^{-1}[\phi] \rangle_{\mathcal{L}_N \otimes \mathcal{L}_N}^V$. The deduction of the asymptotic expansion for linear statistics from the SDE is based on a so-called a priori bound, which we will assume for now, of the following form:

$$ | \langle \phi \rangle_k \otimes_{\mathcal{L}_N} | \leq C \frac{\| \phi \|}{N^{k/2}} \quad (13) $$

for a norm $\|\cdot\|$ that we don't make precise here. Note however, that in the high temperature, one has a to deal with a more complex norm which requires integrability conditions on the functions we apply this bound to. Assuming we know that

$$ \langle \mathcal{D} \circ \Xi^{-1}[\phi] \rangle_{\mathcal{L}_N \otimes \mathcal{L}_N}^V = \frac{\alpha_1(\phi)}{N} + o(N^{-1}) \quad \text{and that} \quad \langle \psi \rangle_{\mathcal{L}_N}^V = o(1), $$

for some $\alpha(\phi) \in \mathbb{R}$, then (12) allows one to obtain the leading order asymptotic for the 1-statistic:

$$ \langle \phi \rangle_{\mathcal{L}_N}^V = N^{-1} \left( P \left\langle \left( \Xi^{-1}[\phi] \right)' \right\rangle_{\mu_V} - \alpha_1(\phi) P + o(1) \right) =: \gamma_1(\phi) N^{-1} + o(N^{-1}). $$

Assuming now that for $n = 2$

$$ \langle \mathcal{D} \circ \Xi^{-1}[\phi] \rangle_{\mathcal{L}_N \otimes \mathcal{L}_N}^V = \sum_{i=1}^{n} \frac{\alpha_i(\phi)}{N^i} + o(N^{-n}), \quad (14) $$

it is not hard to see that one can iteratively derive the expansion of $\langle \phi \rangle_{\mathcal{L}_N}^V$ and get:

$$ \langle \phi \rangle_{\mathcal{L}_N}^V = \sum_{i=1}^{n} \frac{\gamma_i(\phi)}{N^i} + o(N^{-n}), \qquad \gamma_2(\phi) = P \alpha_1 \left( \left( \Xi^{-1}[\phi] \right)' - P \alpha_i(\phi) \right). $$

By the same procedure, one can see that the extraction of the asymptotic expansion up to order $n > 2$ of the 1-linear statistics boils down to extracting the one for the 2-linear statistics.
To achieve that, one needs to investigate the loop equation at level 2, which has the following form for a smooth function $\phi_2$ of 2 variables

$$ \langle \phi_2 \rangle_{\mathcal{L}_N \otimes \mathcal{L}_N}^V = \frac{1}{N} \langle \mathcal{U}[\phi_2] \rangle_{\mathcal{L}_N}^V + \langle \mathcal{V}[\phi_2] \rangle_{\mathcal{L}_N \otimes \mathcal{L}_N}^V + \frac{1}{N} \langle \mathcal{W}[\phi_2] \rangle_{\mathcal{L}_N \otimes \mathcal{L}_N}^V + \frac{1}{N} \langle \mathcal{Y}[\phi_2] \rangle_{\mu_V} $$

with $\mathcal{U}$, $\mathcal{V}$, $\mathcal{W}$ and $\mathcal{Y}$ some operators. From estimate (13) that we assumed at the beginning, we know that

$$ \langle \mathcal{U}[\phi_2] \rangle_{\mathcal{L}_N}^V = O(N^{-1/2}), \quad \langle \mathcal{V}[\phi_2] \rangle_{\mathcal{L}_N \otimes \mathcal{L}_N}^V = O(N^{-3/2}), \quad \langle \mathcal{W}[\phi_2] \rangle_{\mathcal{L}_N \otimes \mathcal{L}_N}^V = O(N^{-1}). $$

It is straightforward to see that only the term $N^{-1} \langle \mathcal{Y}[\phi_2] \rangle_{\mu_V}$ yields a non-negligible contribution to the expansion of $\langle \phi_2 \rangle_{\mathcal{L}_N \otimes \mathcal{L}_N}^V$ at precision $o(N^{-1})$. In order to push it up to $o(N^{-2})$, one needs to obtain the asymptotic expansion for the 3-linear statistics and so on. Each additional order in the asymptotic expansion requires analysing a higher level SDE. Hopefully, each time only a finite number of equations need to be analyzed in order to get all the contributions, and the estimate allows one to neglect all the other terms. Finally, in order to apply the estimate to neglect the remainders, one needs to show that the operators involved in the SDE preserve enough of the regularity of the function they act on, especially for the inverse of the master operator $\Xi^{-1}$. In this setting, one has to obtain way more subtle controls compared to the constant $\beta$-setting. This is due to the fact that (13) involves a more complex norm than just a $L^\infty$-norm. Moreover, finding a manageable integral representation for $\Xi^{-1}[\phi]$ in order to extract controls out of it, is a highly non-trivial step. This makes the proof of the continuity of $\Xi^{-1}[\phi]$ with respect to $\phi$ quite technical. Finding such a form and proving continuity results for $\Xi^{-1}$ is one of the main technical contribution of this article.

When integrating the asymptotic expansion of the 1-linear statistics $\langle \phi \rangle_{L_N}^{V_{G,\phi,t}}$, one needs to justify that the resulting integrals are finite, ie that the integrands are integrable. Justifying that $t \mapsto \rho_{V_{G,\phi,t}}$ is continuous with respect to the uniform convergence norm (of the function and all its derivatives) is enough to conclude this. A transport-based approximation of an analogue of $\rho_{V_{G,\phi,t}}$ was constructed in [LS18] in the case of the 2D Coulomb gas. While for the classical $\beta$-ensembles, it is not hard to show, under some hypotheses on the supports of $V$ and $W$, that

$$ \mu_{tV+(1-t)W} = t\mu_V + (1-t)\mu_W, $$

and deduce the continuity of $t \mapsto \mu_{tV+(1-t)W}$, to analog result in the high temperature regime is much more involved. This is due to the non-linearity of (4).

To show this result, our method is based on an application of the Banach fixed-point theorem to (5). In the model with fixed $\beta$ the analogous step may be done easily because of the linearity, with respect to the equilibrium measure, of the characterizing equation (the analogue (4)). Due to the presence of entropy, $\rho_V$ is the solution of a non-linear integral equation.

## 1.5 Notations and conventions

* Let $X$ be a open set of $\mathbb{R}^p$, we denote by $C^k(X)$ (resp. $L^p(X)$) the space of functions differentiable $k$-times for which the $k$-th derivative is continuous (resp. $p$-th-power integrable functions) on $X$. $C_c^k(X)$ denotes the space of functions of class $k$ on $X$ with compact support. For $p \in [1, +\infty]$, we denote by $L^p(X)$ the usual Lebesgue spaces on $X$ and by $L^p(\mu)$ the Lebesgue spaces with respect to a borelian measure $\mu$ on $\mathbb{R}$. Furthermore, we define $L_0^2(\mu)$ by $\{u \in L^2(\mu), \int_{\mathbb{R}} ud\mu = 0\}$. For a function of several variables $f$, we denote the derivative operator with respect to its $i$-th variable by $\partial_i f$.
* The space of functions $f$ such that $f^{(k)} \in L^\infty(\mathbb{R})$ for all $k = 0, \dots, n$ will be denoted $W_n^\infty(\mathbb{R})$. Its norm is classically $\|f\|_{W_n^\infty(\mathbb{R})} := \max_{k \in \{0, n\}} \|f^{(k)}\|_{L^\infty(\mathbb{R})}$.

* Let $f \in L^2(\mathbb{R})$, we denote by $\mathcal{H}[f]$ the Hilbert transform of $f$ defined by

$$ \mathcal{H}[f](x) := \int_{\mathbb{R}} \frac{f(y)}{y-x} dy $$

where $f$ stands for the Cauchy principal value integral.

* We denote the Fourier transform of $f \in L^1(\mathbb{R}) \cap L^2(\mathbb{R})$ by

$$ \mathcal{F}[f](t) := \int_{\mathbb{R}} f(x)e^{-itx} dx. $$

When $\mu$ is a signed measure over $\mathbb{R}$, we shall denote its Fourier transform by the same symbol $\mathcal{F}[\mu]$.

* The 1/2-norm is defined for any function $f$ which makes this quantity finite

$$ \|f\|_{1/2}^2 := \int_{\mathbb{R}} |t| |\mathcal{F}[f](t)|^2 dt. $$

* We denote by $M_1(\mathbb{R})$ the set of probability measures over $\mathbb{R}$. For $\mu, \mu' \in M_1(\mathbb{R})$ we define the distance (possibly infinite) $D$ by

$$ D(\mu, \mu') := \left( \int_{0}^{+\infty} \frac{1}{t} |\mathcal{F}[\mu - \mu')(t)|^2 dt \right)^{1/2}. \qquad (15) $$

* We define the Sobolev spaces for all $m \ge 0$ by

$$ H^m(\mathbb{R}^n) := \{u \in L^2(\mathbb{R}^n), \|u\|_{H^m(\mathbb{R}^n)} < +\infty\} $$

where

$$ \|u\|_{H^m(\mathbb{R}^n)}^2 := \int_{\mathbb{R}^n} (1 + \|t\|_2)^{2m} |\mathcal{F}[u](t_1, \dots, t_n)|^2 d^n t. $$

Above, $\|\cdot\|_2$ denotes the Euclidean norm on $\mathbb{R}^n$. If $\mu \in M_1(\mathbb{R})$, we also define

$$ H^k(\mu) := \{u \in L^2(\mu), u^{(k)} \in L^2(\mu)\}. $$

**Outline of the paper.** In Section 2, we establish an a priori bound on the $n$-linear statistics that will be crucial in order to analyze the loop equations. To prove this bound, we first prove a concentration inequality for the empirical measure. In Section 3, we establish controls on the operators that appear as building blocks of the loop equations. In Section 4, we prove controls on the so-called master operator. These will play a crucial role in the analysis of the loop equations. We then state the loop equations and establish the large $N$ asymptotic expansion of the linear statistics in Section 5. In Section 6, we establish the continuity of the equilibrium density associated with the interpolation between the Gaussian potential and the potential considered in this paper, and this with respect to the interpolation parameter. Section 7 is dedicated to the expansion of the
partition function and an explicit form for the free energy associated with the Gaussian potential, as well as the interpolation formula. We conclude with Theorem 1.4 thanks to the results shown in Section 6, including Theorem 1.3. This allows us to integrate the asymptotic expansion obtained in Theorem 1.2. We detail in Appendix A some results obtained in [DGM23] upon which this article largely relies. In Appendix B, we prove the continuity and the integrability of the constants that appear in our problem.

**Acknowledgements:** The author wishes to thank Alice Guionnet, Gaultier Lambert and Trinh Khanh Duy for useful suggestions and interesting discussions about this article. I also thank Karol Kozlowski for his valuable advice and his idea for showing the continuity of the equilibrium density with respect to the interpolation.

# 2 A priori bound on the linear statistics

As explained in the introduction, before analyzing the loop equations, one needs a bound that quantifies how small is a function integrated $n$ times against the recentered empirical measure $\mathcal{L}_N := L_N - \mu_V$. Before addressing this, let us recall certain properties enjoyed by $\mu_V$ and the concentration results established in [DGM23].

## 2.1 Equilibrium measure

We recall the definition of the logarithmic potential (or sometimes called Symm's operator) $U^f$ of a function $f: \mathbb{R} \to \mathbb{R}$. When it is defined, the latter is given for all $x \in \mathbb{R}$ by

$$ U^f(x) = - \int_{\mathbb{R}} \log |x - y| f(y) dy. \qquad (16) $$

One can check that $(U^f)' = \mathcal{H}[f]$.

We now describe the regularity of the equilibrium density $\rho_V$ characterized by (4).

**Lemma 2.1** [DGM23][Lemma 2.2]

* The support of $\mu_V$ is $\mathbb{R}$ and there exists a $P$-dependent constant $C_V$ such that for all $x \in \mathbb{R}$,

$$ \rho_V(x) \le C_V (1 + |x|)^{2P} e^{-V(x)}. $$

* The density $\rho_V \in C^\infty(\mathbb{R})$ and it holds

$$ \rho_V' = -(V' + 2P\mathcal{H}[\rho_V])\rho_V, \qquad (17) $$

as well as

$$ \rho_V'' = (-2P\mathcal{H}[\rho_V]' - V'' + V'^2 + 4P^2\mathcal{H}[\rho_V]^2 + 4PV'\mathcal{H}[\rho_V])\rho_V. \qquad (18) $$

## 2.2 Concentration inequality

We now use an idea introduced by [MMS14] and based on a comparison between a configuration $\underline{x} = (x_1, \dots, x_N)$ sampled with $\mathbb{P}_{\underline{x}}^{V,P}$ and a regularized version $\underline{y} = (y_1, \dots, y_N)$, which we describe here.
**Definition 2.2** Let $\underline{x} = (x_1, \dots, x_N) \in \mathbb{R}^N$ and suppose (up to reordering) that $x_1 \le x_2 \le \dots \le x_N$. We define $\underline{y} \in \mathbb{R}^N$ by:

$$y_1 := x_1 \quad \text{and} \quad \forall 0 \le k \le N-1, \quad y_{k+1} := y_k + \max \{x_{k+1} - x_k, e^{-(\log N)^2}\}.$$

We denote by $L_N^{(y)} := \frac{1}{N} \sum_{a=1}^n \delta_{y_a}$ and also define $L_{N,u}^{(y)} := L_N^{(y)} * \mathcal{U}_N$ the convolution between $L_N^{(y)}$ and $\mathcal{U}_N$ the uniform measure on $[0, N^{-2}e^{-(\log N)^2}]$.

Note that the configuration $\underline{y}$ given by the previous definition satisfies $y_{k+1} - y_k \ge e^{-(\log N)^2}$, and $\underline{y}$ is close to $\underline{x}$ in the sense that

$$\sum_{k=1}^{N} |x_k - y_k| \le N^2 e^{-(\log N)^2}. \qquad (19)$$

One can note that we have $|x_k - y_k| = y_k - x_k \le (k-1)e^{-(\log N)^2}$, and we get (19) by summing these inequalities. As in the proof of [DGM23][Theorem 1.5], we obtain a bound on the density:

**Theorem 2.3** For all $N \ge 1$ and $\underline{x} = (x_1, \dots, x_N) \in \mathbb{R}^N$,

$$p_N^{V,P}(\underline{x}) \le \exp \left( -NPD^2 \left( L_{N,u}^{(y)}, \mu_V \right) + K_V + 2P(\log N)^2 \right) \prod_{i=1}^{N} \rho_V(x_i) \quad (20)$$

where $K_V := 2P\|\mathcal{H}[\rho_V]\|_{\infty} + C + P \left| \iint_{\mathbb{R}^2} \log|x-y|d\mu_V(x)d\mu_V(y) \right|$ for some fixed, $V$-independent constant $C$ and with $D$ as given in (15).

Note that we have to keep the dependence on $V$ in all of the constants involved in our problem.

## 2.3 A priori bound on linear statistics

Thanks to the bound given in Theorem 2.3, we can prove the below a priori bound on the linear statistics. This bound is a priori in the sense that it is not optimal, namely, we will show later that for the $n$-linear statistics are $O(N^{-\lceil n/2 \rceil})$ versus $O(N^{-n(1-\epsilon)/2})$ as predicted by the a priori bound. Anyway, this will allow us to neglect, in the loop equations, the terms that are integrated with respect to $\otimes^n \mathcal{L}_N$, where $\mathcal{L}_N := L_N - \mu_V$. The following theorem and its proof are just adaptations of [BGK16][Corollary 3.1.10].

**Theorem 2.4 (A priori bound on linear statistics)** Let $\epsilon > 0$, there exists $C_{n,\epsilon} > 0$ such that for all $f$ in $W_1^{\infty}(\mathbb{R}^n) \cap H^{n/2}(\mathbb{R}^n)$, it holds

$$\left|\langle f \rangle \otimes^n \mathcal{L}_N\right| \le \frac{C_{n,\epsilon} e^{K_V}}{N^{\frac{n}{2}(1-\epsilon)}} \left( \|f\|_{W_1^{\infty}(\mathbb{R}^n)} + \|f\|_{H^{n/2}(\mathbb{R}^n)} \right).$$

where $K_V$ is defined in Theorem 2.3.
**Proof** We use the decomposition $\mathcal{L}_N = (L_N - L_{N,u}^{(y)}) + \mathcal{L}_{N,u}^{(y)}$ where $\mathcal{L}_{N,u}^{(y)} = L_{N,u}^{(y)} - \mu_V$ and obtain:

$$ \langle f \rangle_{\otimes^n \mathcal{L}_N} = \sum_{l=0}^{n-1} \sum_{i_1 < \dots < i_l}^{n} \mathbb{E}_N^{V,P} \left[ \int_{\mathbb{R}^n} f(\xi_1, \dots, \xi_n) \prod_{a=1}^{l} d\mathcal{L}_{N,u}^{(y)}(\xi_{i_a}) \prod_{\substack{a=1 \\ \neq i_1, \dots, i_l}}^{n} d(L_N - L_{N,u}^{(y)})(\xi_a) \right] \\ + \langle f \rangle_{\otimes^n \mathcal{L}_{N,u}^{(y)}}. \quad (21) $$

Since the $x_i$'s are not far from the $y_i$'s, we have the following bound by the mean value theorem and the fact that all the involved measures are probability measures:

$$ \mathbb{E}_N^{V,P} \left[ \int_{\mathbb{R}^n} f(\xi_1, \dots, \xi_n) \prod_{a=1}^{l} d\mathcal{L}_{N,u}^{(y)}(\xi_{i_a}) \prod_{\substack{a=1 \\ \neq i_1, \dots, i_l}}^{n} d(L_N - L_{N,u}^{(y)})(\xi_a) \right] \\ \le C_n \|f\|_{W_1^{\infty}(\mathbb{R}^n)} N e^{-(\log N)^2} \quad (22) $$

for some constant $C_n > 0$ only depending on $n$.

Let's focus now on $\langle f \rangle_{\otimes^n \mathcal{L}_{N,u}^{(y)}}$. We know by Theorem 2.3 that

$$ \mathbb{P}_N^{V,P}(\Omega_N) = e^{KV} O(e^{-cN^\epsilon}) \quad \text{where} \quad \Omega_N := \left\{ \lambda \in \mathbb{R}^N, D^2 \left[ L_{N,u}^{(y)}, \mu_V \right] > \frac{1}{N^{1-\epsilon}} \right\} $$

for some $c > 0$ independent of $V$ and for a remainder controlled $V$-independently. It ensures that:

$$ \left| \langle f \rangle_{\otimes^n \mathcal{L}_{N,u}^{(y)}} \right| \le C e^{KV} e^{-cN^\epsilon} \|f\|_{W_0^\infty(\mathbb{R}^n)} + \mathfrak{R}_N[f] $$

where

$$ \mathfrak{R}_N[f] := \mathbb{E}_N^{V,P} \left[ \mathbf{1}_{\Omega_N^c} \int_{\mathbb{R}^n} f(\xi_1, \dots, \xi_n) d\mathcal{L}_{N,u}^{(y)} \otimes^n (\xi_1, \dots, \xi_n) \right]. $$

By Plancherel formula and Cauchy-Schwarz inequality, one gets:

$$ \begin{aligned} \mathfrak{R}_N[f] &= \mathbb{E}_N^{V,P} \left[ \mathbf{1}_{\Omega_N^c} \int_{\mathbb{R}^n} \mathcal{F}[f](\varphi_1, \dots, \varphi_n) \prod_{a=1}^n \mathcal{F}[\mathcal{L}_{N,u}^{(y)}] (-\varphi_a) \frac{d^n \varphi}{(2\pi)^n} \right] \quad (23) \\ &\le \left( \int_{\mathbb{R}^n} |\mathcal{F}[f](\varphi_1, \dots, \varphi_n)|^2 \prod_{a=1}^n |\varphi_a| \frac{d^n \varphi}{(2\pi)^n} \right)^{1/2} \mathbb{E}_N^{V,P} \left[ \mathbf{1}_{\Omega_N^c} 2^{\frac{n}{2}} D^n \left[ L_{N,u}^{(y)}, \mu_V \right] \right] \\ &\le \frac{2^{\frac{n}{2}}}{N^{\frac{n}{2}(1-\epsilon)}} \left( \int_{\mathbb{R}^n} |\mathcal{F}[f](\varphi_1, \dots, \varphi_n)|^2 \left\{ 1 + \left( \sum_{a=1}^n |\varphi_a|^2 \right)^{1/2} \right\} \frac{d^n \varphi}{(2\pi)^n} \right)^{1/2} \\ &\le 2^{\frac{n}{2}} \frac{\|f\|_{H^{n/2}(\mathbb{R}^n)}}{N^{\frac{n}{2}(1-\epsilon)}} \end{aligned} \quad \square $$

which concludes the proof.
# 3 Properties and control of the operators involved

In this section, we will set some definitions of operators which arise as building blocks of the loop equations. After defining them, we will prove their continuity on appropriate spaces. This will ultimately allow to apply the a priori bounds given in Theorem 2.4.

## 3.1 Definitions

The operators that will appear in the loop equations at level $n \ge 2$ will be constructed via the following extension procedure, allowing one to extend operators acting on $l$ variables into operators acting on $n+l$ variables.

**Definition 3.1 (Extension of operators)** Given an operator $\mathcal{O}$ that acts on functions of one variable and yields a function of $l$ variables, $\phi$ a function of $n$ variables, we define $\mathcal{O}_1$ by:

$$ \mathcal{O}_1[\phi](\xi_1, \dots, \xi_{n+l-1}) = \mathcal{O}[\phi(\cdot, \xi_{l+1}, \dots, \xi_{n+l-1})](\xi_1, \dots, \xi_l) \quad (24) $$

## 3.2 Control on the non-commutative derivative operator

A first example of an operator appearing in the loop equations is the non-commutative derivative (NCD) operator.

**Definition 3.2** Let $f \in C^1(\mathbb{R})$, we define the NCD operator $\mathcal{D}[f]$ by:

$$ \forall x, y \in \mathbb{R}, \mathcal{D}[f](x, y) = \begin{cases} \frac{f(x) - f(y)}{x - y} & \text{if } x \neq y \\ f'(x) & \text{if } x = y \end{cases}. $$

In the following, $p \ge 2$ is fixed.

**Theorem 3.3 (Control for the NCD operator)** Let $n \ge 1$, there exists $C(n) > 0$ such that for all $f \in C^n(\mathbb{R}^{p-1}) \cap H^{n+1}(\mathbb{R}^{p-1})$,

$$ \|\mathcal{D}_1[f]\|_{H^n(\mathbb{R}^p)} \le C(n) \|f\|_{H^{n+1}(\mathbb{R}^{p-1})}. $$

Before showing this inequality, we need to show a general form of the derivatives of $\mathcal{D}_1[f]$.

**Lemma 3.4 (General form for derivatives of $\mathcal{D}_1[f]$)** Let $\underline{m} = (m_1, \dots, m_p) \in \mathbb{N}^p$ satisfy $m_1 \ge m_2$ and $\sum_{i=1}^p m_i \le n$. Let $x_1, \dots, x_p \in \mathbb{R}$ be such that $x_1 \ne x_2$, then one has:

$$ \partial^{\underline{m}} \mathcal{D}_1[f](x_1, \dots, x_n) = \sum_{j=0}^{m_2} C_{m_1, m_2, j} \frac{\left( g^{(j)}(x_2) - \sum_{k=0}^{m_1-j} \frac{g^{(k+j)}(x_1)}{k!} (x_2 - x_1)^k \right)}{(x_2 - x_1)^{m_1+m_2+1-j}} \quad (25) $$

$$ \text{with } C_{m_1, m_2, j} := \binom{m_2}{j} (m_1 + m_2 - j)! (-1)^{m_2-j} \text{ and } g = \partial_2^{m_3} \dots \partial_{p-1}^{m_p} f(\cdot, x_3, \dots, x_p). $$

**Proof** First, it is easy to verify that $\mathcal{D}_1[f] \in C^n(\mathbb{R}^p)$ for $x_1 \ne x_2$. Secondly, when $n \ge 2$, by the Schwarz theorem, the order of the partial derivatives does not matter. It is only the derivatives with respect to $x_1$ and $x_2$ that are non-trivial to compute. Indeed, let $\underline{x} = (x_1, \dots, x_p) \in \mathbb{R}^p$ be such that $x_1 \ne x_2$, then

$$ \partial_3^{m_3} \dots \partial_p^{m_p} \mathcal{D}_1[f](x_1, \dots, x_p) = \frac{g(x_1) - g(x_2)}{x_1 - x_2} $$
with $g := \partial_2^{m_3} \dots \partial_p^{m_p} f(\cdot, x_3, \dots, x_p)$. By applying the Leibniz formula when differentiating $m_1$ times with respect to $x_1$, one gets:

$$ \partial_{x_1}^{m_1} \partial_{x_3}^{m_3} \dots \partial_{x_p}^{m_p} \mathcal{D}_1[f](x_1, \dots, x_p) = \frac{m_1!}{(x_2 - x_1)^{m_1+1}} \left( g(x_2) - \sum_{k=0}^{m_1} \frac{g^{(k)}(x_1)}{k!} (x_2 - x_1)^k \right). $$

Again, we differentiate $m_2$ times with respect to $x_2$ and apply the Leibniz formula to get (25). $\square$

We proved Lemma 3.4 for $m_1 \ge m_2$. Since $\mathcal{D}_1[f]$ is symmetric under the exchange of the two first variables, we can always assume that $m_1 \ge m_2$. The idea of the proof of Theorem 3.3 is to prove separately the $L^2$ control on $\partial^{\mathcal{m}} \mathcal{D}_1[f]$ close to the singularity (the diagonal) and far from it. To do so, we will use the Taylor formula with integral remainder to deal with the singularity and Lemma 3.4 when we are at a fixed distance from the diagonal.

**Proof (of Theorem 3.3)** Let $\underline{m} := (m_1, \dots, m_p) \in \mathbb{N}^p$ be such that $m := \sum_{i=1}^p m_i \le n$. Without loss of generality, we can assume that $m_1 \ge m_2$. Let's show that $\|\partial_1^{m_1} \partial_2^{m_2} \dots \partial_p^{m_p} \mathcal{D}_1[f]\|_{L^2(\mathbb{R}^p)} \le C \|f\|_{H^{m_1+1}(\mathbb{R}^{p-1})}$ with $C > 0$ independent of $f$.

We first show this inequality on the subspace $\{\underline{x} \in \mathbb{R}^n, |x_1 - x_2| \le 1\}$. First note that

$$ \mathcal{D}_1[f](x_1, x_2, \dots, x_p) = \int_0^1 \partial_1 f(x_1 + t(x_2 - x_1), x_3, \dots, x_p) dt $$

an so by differentiating under the integral sign and by Jensen's inequality, we obtain:

$$ \begin{aligned} & |\partial_1^{m_1} \partial_2^{m_2} \dots \partial_p^{m_p} \mathcal{D}_1[f](x_1, x_2, \dots, x_p)|^2 \\ & \le \int_0^1 (1-t)^{2m_1} t^{2m_2} \partial_1^{m_1+m_2+1} \partial_2^{m_2} \dots \partial_{p-1}^{m_p} f(x_1+t(x_2-x_1), x_3, \dots, x_p)^2 dt. \end{aligned} $$

Hence, by integrating with respect to $\underline{x}$, changing $x_2 - x_1$ into $\tilde{x}_2$, and using Fubini, we get:

$$ \begin{aligned} & \int_{\mathbb{R}^n} |\partial_1^{m_1} \partial_2^{m_2} \dots \partial_p^{m_p} \mathcal{D}_1[f](x_1, x_2, \dots, x_p)|^2 \mathbf{1}_{|x_2-x_1|<1} d^n x \\ & \le \int_0^1 dt (1-t)^{2m_1} t^{2m_2} \int_{-1}^1 d\tilde{x}_2 \int_{\mathbb{R}} dx_1 \int_{\mathbb{R}} dx_3 \dots \int_{\mathbb{R}} dx_p \partial_1^{m_1+m_2+1} \partial_2^{m_2} \dots \partial_{p-1}^{m_p} f(x_1+t\tilde{x}_2, x_3, \dots, x_p)^2 \\ & \le C(m_1, m_2) \|\partial_1^{m_1+m_2+1} \partial_2^{m_2} \dots \partial_{p-1}^{m_p} f\|_{L^2(\mathbb{R}^{p-1})}^2 \\ & \le C(m_1, m_2) \|f\|_{H^{m_1+1}(\mathbb{R}^{p-1})}^2. \end{aligned} $$

Now we deal with the subset $\{\underline{x} \in \mathbb{R}^n, |x_1 - x_2| \ge 1\}$. By Jensen's inequality and Lemma 3.4, we get:

$$ \int_{\mathbb{R}} dx_1 \int_{|x_2 - x_1| > 1} dx_2 |\partial_1^{m_1} \partial_2^{m_2} \dots \partial_p^{m_p} \mathcal{D}_1[f](x_1, x_2, \dots, x_p)|^2 \le (m_2 + 1) \sum_{j=0}^{m_2} C_{m_1, m_2, j}^2 \\ \times \int_{\mathbb{R}} dx_1 \int_{|x_2 - x_1| > 1} dx_2 \frac{\left( g^{(j)}(x_2) - \sum_{k=0}^{m_1-j} \frac{g^{(k+j)}(x_1)}{k!} (x_2 - x_1)^k \right)^2}{|x_2 - x_1|^{2m_1+2m_2+2-2j}} $$

with $g = \partial_2^{m_3} \dots \partial_{p-1}^{m_p} f(\cdot, x_3, \dots, x_p)$. Again by Jensen's inequality, we get:
$$
\int_{\mathbb{R}} dx_1 \int_{|x_2-x_1|>1} dx_2 |\partial_1^{m_1} \partial_2^{m_2} \dots \partial_p^{m_p} \mathcal{D}_1[f](x_1, x_2, \dots, x_p)|^2 \le (m_2+1) \sum_{j=0}^{m_2} C_{m_1, m_2, j}^2 (m_1-j+1) \\
\times \int_{\mathbb{R}} dx_1 \int_{|x_2-x_1|>1} dx_2 \frac{g^{(j)}(x_2)^2 + \sum_{k=0}^{m_1-j} \frac{g^{(k+j)}(x_1)^2}{k!^2} (x_2-x_1)^{2k}}{|x_2-x_1|^{2m_1+2m_2+2-2j}}.
$$

For all $j \in [0, m_2]$, the double integral in the last line can be estimated with another constant $C(m_1, m_2)$ depending only on $m_1$ and $m_2$. For that, we use Fubini's theorem:

$$
\begin{align*}
& \int_{\mathbb{R}} dx_1 \int_{|x_2-x_1|>1} dx_2 \frac{g^{(j)}(x_2)^2 + \sum_{k=0}^{m_1-j} \frac{g^{(k+j)}(x_1)^2}{k!^2} (x_2-x_1)^{2k}}{|x_2-x_1|^{2m_1+2m_2+2-2j}} \\
&= \int_{\mathbb{R}} dx_2 g^{(j)}(x_2)^2 \int_{|x_2-x_1|>1} \frac{dx_1}{|x_2-x_1|^{2m_1+2m_2+2-2j}} \\
&\quad + \sum_{k=0}^{m_1-j} \frac{1}{k!^2} \int_{\mathbb{R}} dx_1 g^{(k+j)}(x_1)^2 \int_{|x_2-x_1|>1} \frac{dx_2}{|x_2-x_1|^{2m_1+2m_2+2-2(j+k)}} \\
&\le C(m_1, m_2) \|g\|_{H^{m_1}(\mathbb{R})}^2.
\end{align*}
$$

Hence, after suming over $j$ and changing the constant appropriately, we integrate over $x_3, \dots, x_p$ to
obtain:

$$
\int_{\mathbb{R}^{p-2}} dx_3 \dots dx_p \int_{\mathbb{R}} dx_1 \int_{|x_2-x_1|>1} dx_2 |\partial_1^{m_1} \partial_2^{m_2} \dots \partial_p^{m_p} \mathcal{D}_1[f](x_1, x_2, \dots, x_p)|^2 \\
\le C(m_1, m_2) \sup_{l \in [1, m_1]} \left\| \partial_1^l \dots \partial_{p-1}^{m_p} f \right\|_{L^2(\mathbb{R}^{p-1})}^2 \le C(m_1, m_2) \|f\|_{H^{m_1+1}(\mathbb{R}^{p-1})}^2.
$$

This is enough to conclude.

<figure><img src="image_2.png" /></figure>

Since in Theorem 2.4, the bound on the linear statistic involves the $W_1^\infty(\mathbb{R}^p)$-norm, we state the
following result.

**Proposition 3.5** There exists a $C > 0$ such that for all $f \in C^n(\mathbb{R}^{p-1}) \cap W_n^\infty(\mathbb{R}^{p-1})$,

$$
\| \mathcal{D}_1[f] \|_{W_n^\infty(\mathbb{R}^p)} \le C(n) \|f\|_{W_{n+1}^\infty(\mathbb{R}^{p-1})}.
$$

**Proof** This follows from Lemma 3.4 together with the Taylor formula with integral remainder. $\square$

4 Control on the master operator Ξ

In this section, we study the so-called master operator which will play an essential role in the
following. Indeed, proving continuity of this operator is a crucial step if one wants to analyze the
loop equations.
## 4.1 Definition

We recall the definition of the operator $\mathcal{L}$.

**Definition 4.1** We define, for a sufficiently smooth function $f$, the operator

$$ \mathcal{L}[f] := \Xi[f'] = -\mathcal{A}[f] - 2PW[f] $$

where

$$ \mathcal{A}[f] := -\frac{(f'\rho_V)'}{\rho_V} \quad \text{and} \quad \mathcal{W}[f] := -\mathcal{H}[f'\rho_V] + \int_{\mathbb{R}} \mathcal{H}[f'\rho_V](y)d\mu_V(y). $$

$\mathcal{L}$ is an unbounded operator on the space $\mathbb{H}$ defined in (7). This space is indeed a Hilbert-space by the fact that $\mu_V$ verifies the Poincaré inequality (see assumption (iii)). Its domain is defined by $\mathcal{D}(\mathcal{L}) = \mathcal{D}(\mathcal{A}) := \{u \in \mathbb{H}, \mathcal{A}[u] \in \mathbb{H}\}$ by [DGM23][Theorem 6.7]. We show that the functions belonging to this set are smooth.

**Lemma 4.2 (Regularity of the inverse of the derivative)** [DGM23][Theorem 7.1] Let $v \in \mathcal{D}(\mathcal{L})$, then $v' \in \mathcal{C}^1(\mathbb{R})$.

For the next theorem, we recall that $\mathcal{A}: \mathcal{D}(\mathcal{A}) \to \mathbb{H}$ is a diagonalizable operator with positive countable spectrum. We denote by $\lambda_1(\mathcal{A}) > 0$ its smallest eigenvalue. This quantity has a role in our problem since for all $f \in \mathbb{H}$, $\|\mathcal{L}^{-1}[f]\|_{\mathbb{H}} \le \lambda_1(\mathcal{A})^{-1/2}\|f\|_{\mathbb{H}}$ see [DGM23][Theorem 6.7].

**Theorem 4.3 (Inversion of the master operator)** $\Xi: \mathfrak{D}(\Xi) \to \mathbb{H}$ is invertible, of inverse defined for all $g \in \mathbb{H}$ by:

$$ \Xi^{-1}[g] := (\mathcal{L}^{-1}[g])' $$

where $\mathfrak{D}(\Xi) := \{f \in \mathcal{C}^0(\mathbb{R}), \exists v \in \mathfrak{D}(\mathcal{L}), f = v'\}$. Furthermore for all $f \in \mathbb{H}$,

$$ \|\Xi^{-1}[f]\|_{L^2(\mu_V)} \le C_{\mathcal{L}} \|f'\|_{L^2(\mu_V)} \quad (26) $$

where $C_{\mathcal{L}} := \lambda_1(\mathcal{A})^{-1/2}$.

**Proof** To prove that $\Xi$ is invertible on $\mathfrak{D}(\Xi)$, the only thing to prove is that for all $v \in \mathfrak{D}(\mathcal{L})$, $v' \in \mathcal{C}^0(\mathbb{R})$ which is true by Lemma 4.2. The estimate comes from the fact that given $f \in \mathbb{H}$, one has $\Xi^{-1}[f] = (\mathcal{L}^{-1}[f])'$. Then

$$ \|\Xi^{-1}[f]\|_{L^2(\mu_V)} = \|\mathcal{L}^{-1}[f]\|_{\mathbb{H}} \le C_{\mathcal{L}} \|f\|_{\mathbb{H}}. \quad \square $$

The crucial step when one wants to analyze the loop equations, is to obtain controls on the master operator which we will show in this section. These bounds will allow us to apply the bound obtained in Theorem 2.4 to functions like $\Xi^{-1}[\phi]$.

## 4.2 Preliminaries

We define an operator $\mathcal{O}$ whose iterations will appear in the derivatives of the inverse of the master operator (which exists because of Lemma 4.8).

**Definition 4.4** Let $\mathcal{O}$ be the operator defined on smooth enough functions by:

$$ \mathcal{O}[f](x) := \left(\frac{f\rho_V}{\rho_V'}\right)'(x) \quad x - a-e \quad (27) $$
In order to give a more precise description of $\mathcal{O}^k$, which will allow us to analyse its asymptotics at infinity, we need the following definition.

**Definition 4.5 (Differential degree)** Let $f$ be a function of one variable defined on $\mathbb{R}$ differentiable $n$ times, we define the differential degree denoted by $d_{\partial}^f$ with respect to $f$ by

$$d_{\partial}^{f} \left( \prod_{k=0}^{n} (f^{(k)})^{\alpha_k} \right) := \sum_{k=0}^{n} k \alpha_k$$

For example the differential degree with respect to $f$ of $(f')^2$ and $f''f$ is 2, while $d_{\partial}^f ((f^{(3)})^2) = 6$.

**Remark 4.6** For example, with $\alpha := \frac{\rho_V}{\rho'_V}$, it holds that:

$$\bullet \mathcal{O}[f] = \alpha'f + \alpha f'$$

$$\bullet \mathcal{O}^2[f] = (\alpha\alpha')'f + 3\alpha\alpha'f' + \alpha^2 f''$$

$$\bullet \mathcal{O}^3[f] = (\alpha(\alpha\alpha'))'f + (4\alpha^2\alpha'' + 7\alpha\alpha'^2)f' + (6\alpha'\alpha^2)f'' + \alpha^3 f^{(3)}$$

Using the notion of differential degree, we are now able to state the next theorem.

**Theorem 4.7** Let $k \ge 1$, $f \in \mathcal{C}^k(\mathbb{R})$, there exists a family of polynomials $(P_a^k)_{0 \le a \le k}$ such that

$$\mathcal{O}^k[f] = \sum_{j=0}^{k} f^{(k-j)} P_j^k(\alpha, \dots, \alpha^{(j)}), \quad \text{with} \quad \alpha := \frac{\rho_V}{\rho'_V} \qquad (28)$$

In fact, $P_j^k(\alpha, \dots, \alpha^{(j)})$, $j \in [0, k]$, is the unique homogeneous polynomial in $j+1$ variables, with differential degree with respect to $\alpha$ equal to $j$, degree $k$ and with coefficients independent of $V$ satisfying the following recurrence relations:

$$\bullet P_0^{k+1}(\alpha) = \alpha P_0^k(\alpha) = \alpha^{k+1}$$

$$\bullet \forall j \in [1, k], P_j^{k+1}(\alpha, \dots, \alpha^{(j)}) = (\alpha P_{j-1}^k(\alpha, \dots, \alpha^{(j-1)}))' + \alpha P_j^k(\alpha, \dots, \alpha^{(j)})$$

$$\bullet P_{k+1}^{k+1}(\alpha, \dots, \alpha^{(k+1)}) = (\alpha P_k^k(\alpha, \dots, \alpha^{(k)}))' = ((\alpha'\alpha)' \dots \alpha)'$$

**Proof** Let's prove it by induction. For $k=1$, $\mathcal{O}[f] = \alpha'f + \alpha f'$ and so by setting $P_0^1(\alpha) = \alpha$, which is homogeneous, of degree 1 and of differential degree 0, and $P_1^1(\alpha, \alpha') = \alpha'$ which is of degree 1 and differential degree 1, this proves the claim. Suppose that (28) holds at rank $k \in \mathbb{N}^*$, then

$$\begin{align*}
\mathcal{O}^{k+1}[f] &= (\alpha \mathcal{O}^k[f])' = \sum_{j=0}^{k} f^{(k-j)} [\alpha P_j^k(\alpha, \dots, \alpha^{(j)})]' + \sum_{j=0}^{k} f^{(k-j+1)} \alpha P_j^k(\alpha, \dots, \alpha^{(j)}) \\
&= [\alpha P_k^k(\alpha, \dots, \alpha^{(k)})]' f + \sum_{j=0}^{k-1} f^{(k-j)} \left\{ [\alpha P_j^k(\alpha, \dots, \alpha^{(j)})]' + \alpha P_{j+1}^k(\alpha, \dots, \alpha^{(j+1)}) \right\} \\
&\qquad + \alpha P_0^k(\alpha) f^{(k+1)}
\end{align*}$$

Hence by setting $P_0^{k+1}(\alpha) := (\alpha P_0^k(\alpha))'$, $P_{k+1}^{k+1}(\alpha, \dots, \alpha^{(k+1)}) := (\alpha P_k^k(\alpha, \dots, \alpha^{(k)}))'$ and for all $j \in [0, k-1]$, $P_{j+1}^{k+1}(\alpha, \dots, \alpha^{(j+1)}) := (\alpha P_j^k(\alpha, \dots, \alpha^{(j)}))' + \alpha P_j^k(\alpha, \dots, \alpha^{(j)})$, we obtain the desired form of (28) and the recurrence relations. It remains to check that the homogeneity and degree conditions hold at rank $k+1$. This follows from the recurrence relations for the $P_j^k$'s. $\square$
## 4.3 Closed form for $\Xi^{-1}$

Before showing a closed form for the derivatives of $\Xi^{-1}[f]$ and their $L^2$ properties, we first prove that, if $f$ is sufficiently smooth, they indeed exist.

**Lemma 4.8 (Regularity of the inverse)** Let $f \in H$ such that $f\rho_V \in H^n(\mathbb{R})$ with $n \ge 2$, then $\rho_V\Xi^{-1}[f] \in H^{n+1}(\mathbb{R})$. Furthermore if $f \in H \cap C^n(\mathbb{R})$ is such that $f\rho_V \in H^n(\mathbb{R})$, then one also has $\Xi^{-1}[f] \in C^{n+1}(\mathbb{R})$.

Note that the last condition is verified whenever $f$ and its derivatives are continuous and grow slower than $e^{-V}$ at infinity. The proof uses the operators $\mathcal{L}$ and $\mathcal{A}$ introduced in Definition 4.1.

**Proof** We recall that $\Xi^{-1}[f] = (\mathcal{L}^{-1}[f])'$. When $f \in H$, we know that $\rho_V(\mathcal{L}^{-1}[f])' \in H^2(\mathbb{R})$. This is because $\mathcal{L}^{-1}[f] \in \mathcal{D}(\mathcal{L}) \subset \{u \in H, \mathcal{A}[u] \in H\}$ and $(\rho_V(\mathcal{L}^{-1}[f])')' = \rho_V\mathcal{A}[\mathcal{L}^{-1}[f]] \in H^1(\mathbb{R})$. We want to show that $\rho_V(\mathcal{L}^{-1}[f])'' \in H^n(\mathbb{R})$, let's show first that $\mathcal{H}[\rho_V(\mathcal{L}^{-1}[f])'] \in H^n(\mathbb{R})$. First observe that

$$ \rho_V \mathcal{A} \mathcal{L}^{-1}[f] = -\rho_V f + 2P \rho_V \mathcal{H} [\rho_V (\mathcal{L}^{-1}[f])'] - 2P \rho_V \int_{\mathbb{R}} \mathcal{H} [\rho_V (\mathcal{L}^{-1}[f])'] (y) dy $$

Hence, since $\rho_V(\mathcal{L}^{-1}[f])' \in H^2(\mathbb{R})$, so is $\mathcal{H}[\rho_V(\mathcal{L}^{-1}[f])']$. Moreover $2P\rho_V \int_{\mathbb{R}} \mathcal{H}[\rho_V(\mathcal{L}^{-1}[f])'] (y) dy$ clearly belongs to $H^n(\mathbb{R})$ for all $n \in \mathbb{N}$ hence $\rho_V\mathcal{A} \circ \mathcal{L}^{-1}[f] = (\rho_V(\mathcal{L}^{-1}[f])')' \in H^2(\mathbb{R})$ and hence $\rho_V(\mathcal{L}^{-1}[f])' \in H^3(\mathbb{R})$. By induction, this shows that $(\mathcal{L}^{-1}[f])' = \Xi^{-1}[f] \in \frac{1}{\rho_V}H^{n+1}(\mathbb{R}) \subset C^n(\mathbb{R})$ by Sobolev-Hölder embedding theorem and hence that $\mathcal{H}[\rho_V(\mathcal{L}^{-1}[f])'] \in H^{n+1}(\mathbb{R})$. Since

$$ (\mathcal{L}^{-1}[f])'' = f - \frac{\rho_V'}{\rho_V} (\mathcal{L}^{-1}[f])' - 2P \left( \mathcal{H} [\rho_V (\mathcal{L}^{-1}[f])'] - \int_{\mathbb{R}} \mathcal{H} [\rho_V (\mathcal{L}^{-1}[f])'] (y) d\mu_V(y) \right) \quad (29) $$

and that $\frac{\rho_V'}{\rho_V} \in C^\infty(\mathbb{R})$, we can then conclude that, under the assumption that $f \in C^n(\mathbb{R})$, $(\Xi^{-1}[f])' = \mathcal{L}^{-1}[f] \in C^n(\mathbb{R})$, hence $\Xi^{-1}[f] \in C^{n+1}(\mathbb{R})$. $\square$

**Lemma 4.9** There exists $M_V > 0$ such that $\forall |x| \ge M_V$, $|\frac{\rho_V'}{\rho_V}(x)| \ge 1$.

**Proof** From Lemma A.2, $\mathcal{H}[\rho_V]$ is bounded and by assumption (ii), $V'(x)$ goes to infinity, the conclusion follows from the fact that $\frac{\rho_V'}{\rho_V}(x) = -V'(x) - 2P\mathcal{H}[\rho_V]$. $\square$

We are now able to prove that a closed form holds for the derivatives of $\Xi^{-1}$. The idea is to use the resolvant formula which gives that for all $f \in H$,

$$ \mathcal{L}^{-1}[f] = -\mathcal{A}^{-1} [f + 2PW \circ \mathcal{L}^{-1}[f]] \quad (30) $$

and for all $x \in \mathbb{R}$,

$$ \mathcal{A}^{-1}[f](x) = \frac{1}{\rho_V(x)} \int_x^{\pm\infty} f(t) \rho_V(t) dt. \quad (31) $$
It doesn't matter if one chooses $+\infty$ or $-\infty$ in (31) since $\int_{\mathbb{R}} f(t)\rho_V(t)dt = 0$ but it will be convenient to make the choice sgn($x$)$\infty$ for reasons that will appear further. Before establishing the continuity for $\Xi^{-1}$, we need to introduce an operator $\mathcal{X}$ that takes a function in $\frac{1}{\rho_V}H^n(\mathbb{R})$ and produces one belonging to $\frac{1}{\rho_V}H^n(\mathbb{R}) \cap \mathcal{H}$ by means of a recentering.

**Definition 4.10** Let $\phi \in \frac{1}{\rho_V}H^n(\mathbb{R})$, we define the operator $\mathcal{X}$ by

$$ \mathcal{X}[\phi](\xi) = \phi(\xi) - \int_{\mathbb{R}} \phi(t)d\mu_V(t). $$

For any $\phi \in \frac{1}{\rho_V}H^n(\mathbb{R})$, it is clear that $\rho_V\mathcal{X}[\phi] \in H^n(\mathbb{R})$ ie $\mathcal{X}[\phi] \in \frac{1}{\rho_V}H^n(\mathbb{R})$. We denote by

$$ \widetilde{\Xi}_1 := \Xi_1 \circ \mathcal{X}_1, \qquad \widetilde{\Xi}_1^{-1} := \Xi_1 \circ \mathcal{X}_1 $$

and, given a general operator $\mathcal{U}$, we adopt the notation $\tilde{\mathcal{U}}$ for the operator $\mathcal{U} \circ \mathcal{X}$.

**Theorem 4.11** Let $f \in C^n(\mathbb{R}) \cap \left(\frac{1}{\rho_V}H^n(\mathbb{R})\right)$, for all $|x| > M_V$ with $M_V$ given in Lemma 4.9, for all $k \in [1, n+1]$ it holds that

$$ (\widetilde{\Xi^{-1}[f]})^{(k)} = \sum_{a=0}^{k-1} Q_a^k(\theta, \dots, \theta^{(a)}) \beta_{k-a}, \quad \text{where} \quad \theta := \frac{\rho_V'}{\rho_V} \qquad (32) $$

with the convention that $\widetilde{\Xi^{-1}[f]} =: \beta_0$. The $\beta_i$'s are defined, for all $|x| > M_V$, for all $i \in [1, k]$, by:

$$ \beta_i(x) := \frac{-1}{\rho_V(x)} \int_{x}^{\operatorname{sgn}(x)\infty} dt \rho_V(t) \mathcal{O}^i \left[ \mathcal{X}[f] + 2PW \circ \widetilde{\mathcal{L}^{-1}[f]} \right] (t) $$

(see (27)). Above $Q_a^k$ denotes the unique homogeneous polynomial in $a+1$ variables with degree $k-a$, with differential degree with respect to $\theta$ $a$ and with coefficients independent of $V$ satisfying the following induction relations:

$$ Q_0^{k+1}(\theta) = \theta Q_0^k(\theta) = \theta^k \qquad (33) $$

$$ \forall a \in [1, k-1], Q_a^{k+1}(\theta, \dots, \theta^{(a)}) = \theta Q_a^k(\theta, \dots, \theta^{(a)}) + Q_{a-1}^k(\theta, \dots, \theta^{(a)})' \qquad (34) $$

$$ Q_k^{k+1}(\theta, \dots, \theta^{(k)}) = Q_{k-1}^k(\theta, \dots, \theta^{(k-1)})' = \theta^{(k-1)} \qquad (35) $$

**Proof** We prove this statement by induction. For $k=1$, by (30) and (31), by setting

$$ g := -\mathcal{X}[f] - 2PW \circ \widetilde{\mathcal{L}^{-1}[f]}, $$

we get for all $x \in \mathbb{R}$,

$$ \widetilde{\Xi^{-1}[f]}(x) = (\widetilde{\mathcal{L}^{-1}[f]})'(x) = (\mathcal{A}^{-1}[g])'(x) = \frac{1}{\rho_V(x)} \int_{x}^{\operatorname{sgn}(x)\infty} dt \rho_V(t) g(t) = \beta_0(x). \qquad (36) $$
For $k=2$, differentiating again, which is allowed by Lemma 4.8, we get for $|x|$ large enough:

$$ (\widetilde{\mathcal{L}^{-1}[f]})''(x) = (\mathcal{A}^{-1}[g])''(x) = -g(x) - \frac{\rho'_{V}}{\rho_{V}}(x) (\mathcal{A}^{-1}[g])'(x). $$

After performing an integration by parts in the last integral, we obtain

$$ (\widetilde{\Xi^{-1}[f]})'(x) = (\widetilde{\mathcal{L}^{-1}[f]})''(x) = \frac{\rho'_{V}}{\rho_{V}^{2}}(x) \int_{x}^{\operatorname{sgn}(x)\infty} dt \rho_{V}(t) \left( g \frac{\rho_{V}}{\rho'_{V}} \right)'(t) = (\theta\beta_{1})(x). $$

By defining $Q_0^1(\theta) := \theta$, it is readily seen that its degree is 1 and its differential degree with respect to $\theta$ is 0. Let $k \in [2, n]$ and suppose that (32) is true at rank $k$, then by differentiating we get:

$$ (\widetilde{\Xi^{-1}[f]})^{(k+1)} = \sum_{a=0}^{k-1} Q_a^k(\theta, \dots, \theta^{(a)}) \beta_{k-a}' + Q_a^k(\theta, \dots, \theta^{(a)})' \beta_{k-a}. \quad (37) $$

First, let $i \in [1, k]$ and $|x|$ be large enough,

$$ \beta_i'(x) = -\mathcal{O}^i[g](x) - \frac{\rho'_{V}}{\rho_{V}^{2}}(x) \int_{x}^{\operatorname{sgn}(x)\infty} dt \rho_{V}(t) \mathcal{O}^i[g](t) = \frac{\rho'_{V}}{\rho_{V}^{2}}(x) \int_{x}^{\operatorname{sgn}(x)\infty} dt \rho_{V}(t) \mathcal{O}^{i+1}[g](t) = (\theta\beta_{i+1})(x). $$

The second equality follows from an integration by parts and the fact that $\frac{\rho'_{V}}{\rho_{V}^{2}}\rho_{V}\frac{\rho_{V}}{\rho'_{V}}\mathcal{O}^i[g]$ goes to zero at infinity. Hence (37) becomes

$$ \begin{aligned} (\widetilde{\Xi^{-1}[f]})^{(k+1)} &= \sum_{a=0}^{k-1} \theta Q_a^k(\theta, \dots, \theta^{(a)}) \beta_{k+1-a} + Q_a^k(\theta, \dots, \theta^{(a)})' \beta_{k-a} \\ &= \theta Q_0^k(\theta) \beta_{k+1} + \sum_{a=1}^{k-1} \left( \theta Q_a^k(\theta, \dots, \theta^{(a)}) + Q_{a-1}^k(\theta, \dots, \theta^{(a-1)})' \right) \beta_{k+1-a} \\ &\quad + Q_{k-1}^k(\theta, \dots, \theta^{(k-1)})' \beta_1. \end{aligned} $$

By the definitions of $(Q_a^{k+1})_a$, it is clear that (32) is true at rank $k+1$. The fact that $Q_a^{k+1}$ are homogeneous and have degree $k-a$ and differential degree $a$ can be checked directly from the induction relations (33), (34), (35). $\square$

**Remark 4.12** When $V(x) = x^m$ with $m$ even, it can be checked from (17) that for every $a \in [0, n-2]$, $Q_a^n(\theta, \dots, \theta^{(a)})$ is of the form $c_{a,n}x^{m(n-1-a)-(n-1)}+T_a^n(x)+R_a^n(x, \mathcal{H}[\rho_V], \dots, \mathcal{H}[\rho_V^{(a)}])(x)$ where $c_{a,n}$ is a real number, $T_a^n$ is polynomial of degree strictly lower than $m(n-1-a)-(n-1)$ and $R_a^n$ is also a polynomial of degree greater than 1. Since all these Hilbert transform vanish at infinity, such a polynomial expression goes to zero at infinity. This decomposition holds as long as the degree of the monomial is non-negative, otherwise it is zero. We give the first decompositions for $(\widetilde{\Xi^{-1}[f]})^{(k)}$ for $k \in [0, 3]$:

$$ \widetilde{\Xi^{-1}[f]} = \beta_0, \qquad (\widetilde{\Xi^{-1}[f]})' = \theta\beta_1, \qquad (\widetilde{\Xi^{-1}[f]})'' = \theta'\beta_1 + \theta^2\beta_2, $$
and

$$ (\tilde{\Xi}^{-1}[f])^{(3)} = \theta''\beta_1 + (\theta\theta' + (\theta^2)')\beta_2 + \theta^3\beta_3. $$

With the choice of potential $V(x) = x^m$ with $m$ even, choosing a bounded function $f$ with bounded derivatives at all orders and integrating by parts, it holds that for all $k \ge 0$, there exists $\gamma_0^{(k)}, \dots, \gamma_k^{(k)} \in \mathbb{R}$,

$$ |\beta_k(x)| \underset{|x| \to \infty}{\sim} \frac{\mathcal{O}^k[g](x)}{x^{m-1}} \underset{|x| \to \infty}{\sim} \frac{1}{x^{m-1}} \sum_{j=0}^k g^{(j)}(x) \left( \frac{\gamma_j^{(k)}}{x^{km-j}} + \underset{|x| \to \infty}{o}\left(\frac{1}{x^{km-j}}\right) \right). $$

When $V(x) = \cosh(\alpha x)$, by the same computation, we get for different $\gamma_j^{(k)}$

$$ |\beta_k(x)| \underset{|x| \to \infty}{\sim} e^{-\alpha|x|} \sum_{j=0}^{k} \gamma_j^{(k)} g^{(j)}(x) \left( e^{-k\alpha|x|} + \underset{|x| \to \infty}{o}\left(e^{-k\alpha|x|}\right) \right). $$

## 4.4 Controls on the inverse of the master operator

Since we are going to use the polynomials, $P_j^k$ and $Q_j^k$ defined previously in Theorem 4.7 and 4.11, a lot in our estimates on $\tilde{\Xi}_1^{-1}$, we first need the following lemma. With $\alpha = \frac{\rho_V}{\rho'_V}\theta^{-1}$, the following result holds.

**Lemma 4.13** For all $k \ge 1$, for all $j \in [0, k]$

$$ (i) \quad P_j^k(\alpha, \dots, \alpha^{(j)})(x) = \mathcal{O}\left(\frac{V'(x)^{-k}}{|x|^{\alpha}}\right), $$

$$ (ii) \quad P_j^k(\alpha, \dots, \alpha^{(j)})'(x) = \mathcal{O}\left(\frac{V'(x)^{-k}}{|x|^{\alpha}}\right), $$

$$ (iii) \quad Q_j^k(\theta, \dots, \theta^{(j)})(x) = \mathcal{O}\left(\frac{V'(x)^{k-j}}{|x|^{\alpha}}\right). $$

**Proof** For $i$), by the Faà di Bruno's formula, for all $n \ge 0$,

$$ \alpha^{(n)} = \left( \frac{-1}{V' + 2PH[\rho_V]} \right)^{(n)} = - \sum_{\lambda \vdash n} \frac{(-1)^{|\lambda|} |\lambda|!}{(V' + 2PH[\rho_V])^{|\lambda|+1}} \prod_{i=1}^{n} \frac{(V^{(i+1)} + 2PH[\rho_V]^{(i)})^{\lambda_i}}{\lambda_i! (i!)^{\lambda_i}}. $$

where the sum is over $\lambda := (\lambda_1, \dots, \lambda_n)$ such that $\sum_{i=1}^n i\lambda_i = n$. From assumption $v)$ and Lemma A.2, we see that

$$ |\alpha^{(n)}(x)| \le \sum_{\lambda \vdash n} C_{\lambda} \mathcal{O}\left(\frac{V'(x)^{-1}}{|x|^{\alpha}}\right) = \mathcal{O}\left(\frac{V'(x)^{-1}}{|x|^{\alpha}}\right). $$

Hence $P_j^k$, as a homogeneous polynomial in $(\alpha, \dots, \alpha^{(j)})$ of degree $k$, is a $\mathcal{O}\left(\frac{V'(x)^{-k}}{|x|^{\alpha}}\right)$.

For the point $ii)$, one has to notice that for each monomial

$$ A_n := \left[ \prod_{i=1}^{n} (\alpha^{(i)})^{l_i} \right]' = \sum_{j=1}^{n} l_j \alpha^{(j+1)} (\alpha^{(j)})^{l_j-1} \prod_{i \ne j}^{n} (\alpha^{(i)})^{l_i}. $$
But, we have proven that for all $i \in \mathbb{N}$, $\alpha^{(i)} = \mathcal{O}_{|x| \to \infty} (V'(x)^{-1})$, so by denoting $l := \sum_{i=1}^n l_i = \deg(A_n)$, $A_n(x) = \mathcal{O}_{|x| \to \infty} (V'(x)^{-l})$. Therefore, any homogeneous polynomial of degree $k$ such as $P_j^k$ in the variables $(\alpha, \dots, \alpha^{(j)})$ is a $\mathcal{O}_{|x| \to \infty} (V'(x)^{-k})$.

Finally for the point iii), it is clear that for all $j \ge 0$,

$$ \theta^{(j)}(x) = V^{(j+1)}(x) - 2PH[\rho_V]^{(j)}(x) = \mathcal{O}_{|x| \to \infty} (V'(x)). $$

Thus $Q_j^{(k)}(\theta, \dots, \theta^{(j)})$ as a homogeneous polynomial of degree $k-j$, is a $\mathcal{O}_{|x| \to \infty} (V'(x)^{k-j})$. $\square$

**Theorem 4.14 (H$^n$(R)-continuity of $\tilde{\Xi}_1^{-1}$)** There exists a constant $C(\tilde{\Xi}_1^{-1}, H^n) > 0$ depending only on $n$ and $V$ such that for all $f \in H^{n+1}(\mathbb{R}^p)$,

$$ \|\tilde{\Xi}_1^{-1}[f]\|_{H^n(\mathbb{R}^p)} \le C(\tilde{\Xi}_1^{-1}, H^n) \|f\|_{H^{n+1}(\mathbb{R}^p)}. $$

Under the choice of potential $V_{\phi,t}$ defined in Theorem 1.3, for $\phi \in \cap_{k \ge 0} H^k(\mathbb{R})$ the map $t \in [0,1] \mapsto C(\tilde{\Xi}_1^{-1}, H^n)$ is continuous.

The idea of the proof is to use the closed form for the operator $\mathcal{O}^k$ defined in 4.4, found in Theorem 4.7 and inject it in the $\beta_k$ which appear in the closed form for $\tilde{\Xi}^{-1}(n)$ in Theorem 4.11.

**Proof** Let $m \le n$ and $(m_1, \dots, m_p) \in \mathbb{N}^p$ be such that $\sum_{i=1}^p m_i = m$. Let $x_2, \dots, x_p \in \mathbb{R}^{p-1}$ be fixed, we define $h: x_1 \mapsto \partial_2^{m_2} \dots \partial_p^{m_p} f(x_1, \dots, x_p)$ and $g = -\mathcal{X}[h] - 2PW \circ \mathcal{L}^{-1}[h]$. With these notations,

$$ \partial^{\underline{m}} \tilde{\Xi}_1^{-1}[f](x_1, \dots, x_p) = \tilde{\Xi}^{-1}[h]^{(m_1)}(x_1). $$

We can then apply Theorem 4.11 and Theorem 4.7, so for $|x_1|$ large enough, we get

$$ \begin{aligned} \widetilde{\Xi}^{-1}[h]^{(m_1)}(x_1) &= \sum_{a=0}^{m_1-1} Q_a^{m_1}(\theta, \dots, \theta^{(a)}) (x_1) \beta_{m_1-a}(x_1) \\ &= \sum_{a=0}^{m_1-1} Q_a^{m_1}(\theta, \dots, \theta^{(a)}) (x_1) \frac{1}{\rho_V(x_1)} \int_{x_1}^{\operatorname{sgn}(x_1)\infty} dt \rho_V(t) \mathcal{O}^{m_1-a}[g](t) \\ &= \sum_{a=0}^{m_1-1} \sum_{b=0}^{m_1-a} Q_a^{m_1}(\theta, \dots, \theta^{(a)}) (x_1) \frac{1}{\rho_V(x_1)} \int_{x_1}^{\operatorname{sgn}(x_1)\infty} dt \rho_V(t) g^{(m_1-a-b)}(t) P_b^{m_1-a}(\alpha, \dots, \alpha^{(b)})(t). \end{aligned} \quad (38) $$

Moreover an integration by parts yields:

$$ \begin{aligned} \widetilde{\Xi}^{-1}[h]^{(m_1)}(x_1) &= \sum_{a=0}^{m_1-1} \sum_{b=0}^{m_1-a} Q_a^{m_1}(\theta, \dots, \theta^{(a)}) (x_1) \left( -g^{(m_1-a-b)}(x_1) \alpha(x_1) P_b^{m_1-a}(\alpha, \dots, \alpha^{(b)})(x_1) \right. \\ & \qquad \left. + \frac{1}{\rho_V(x_1)} \int_{x_1}^{\operatorname{sgn}(x_1)\infty} dt \rho_V(t) \left[ g^{(m_1-a-b+1)}(t) \alpha(t) P_b^{m_1-a}(\alpha, \dots, \alpha^{(b)})(t) \right. \right. \\ & \qquad \left. \left. + g^{(m_1-a-b)}(t) \left[ \alpha P_b^{m_1-a}(\alpha, \dots, \alpha^{(b)}) \right]'(t) \right] \right). \end{aligned} \quad (39) $$
We now use $M_V > 0$ from Lemma 4.9 so that $P_b^a(\alpha, \dots, \alpha^{(b)})$ that appear above are well-defined on $[-M_V, M_V]^c$ ie don't have any singularity. Hence by integrating with respect to $x_1$, $(\Xi^{-1}[h]^{(m_1)})^2$ on $[M_V, +\infty[$, we get by Jensen's inequality for a constant $C(m_1) > 0$ depending only on $m_1$

$$
\begin{aligned}
& \int_{M_V}^{+\infty} dx_1 (\Xi^{-1}[h]^{(m_1)}(x_1))^2 \\
& \le C(m_1) \sum_{a=0}^{m_1-1} \sum_{b=0}^{m_1-a} \int_M dx_1 Q_a^{m_1}(\theta, \dots, \theta^{(a)})(x_1)^2 \left\{ g^{(m_1-a-b)}(x_1)^2 \alpha(x_1)^2 P_b^{m_1-a}(\alpha, \dots, \alpha^{(b)})(x_1)^2 \right. \\
& \qquad \left. + \frac{1}{\rho_V(x_1)^2} \left[ \int_{x_1}^{\operatorname{sgn}(x_1)\infty} dt \rho_V(t) g^{(m_1-a-b+1)}(t) \alpha(t) P_b^{m_1-a}(\alpha, \dots, \alpha^{(b)})(t) \right]^2 \right. \\
& \qquad \left. + \frac{1}{\rho_V(x_1)^2} \left[ \int_{x_1}^{\operatorname{sgn}(x_1)\infty} dt \rho_V(t) g^{(m_1-a-b)}(t) \left[ \alpha P_b^{m_1-a}(\alpha, \dots, \alpha^{(b)}) \right]'(t) \right]^2 \right\}.
\end{aligned}
$$

We want to bound this expression by $\|g\|_{H^{m_1+1}(\mathbb{R})}^2$, but since $g = -h + 2PH[\rho_V\Xi^{-1}[h]] + c$, where $c = \int_{\mathbb{R}} h(y)d\mu_V(y) - 2P\int_{\mathbb{R}} H[\rho_V\Xi^{-1}[h]](y)d\mu_V(y)$, the constant terms will fail to be in $L^2(\mathbb{R})$. We thus have to treat these terms separately. In the previous sum, $g$ is differentiated everywhere except in the term $b = m_1 - a$ so this is the only value of $b$ where we have to deal with $c$. By defining:

$$f_{n_1, a, b}^{(1), V} : x \mapsto Q_a^{n_1} (\theta, \dots, \theta^{(a)})(x) \alpha(x) P_b^{n_1 - a} (\alpha, \dots, \alpha^{(b)})(x) \quad (40)$$

$$f_{n_1, a, b}^{(2), V} : x \mapsto \frac{Q_a^{n_1} (\theta, \dots, \theta^{(a)})(x)}{\rho_V(x)} \int_x^{\operatorname{sgn}(x)\infty} dt \rho_V(t) \left[ \alpha P_b^{n_1-a} (\alpha, \dots, \alpha^{(b)}) \right]'(t), \quad (41)$$

by using Jensen's inequality and inequality (26) we get:

$$
\begin{aligned}
& \mathfrak{c}^2 C(m_1) \sum_{a=0}^{m_1-1} \int_{M_V}^{+\infty} dx_1 \left[ f_{m_1, a, m_1-a}^{(1),V}(x_1)^2 + f_{m_1, a, m_1-a}^{(2),V}(x_1)^2 \right] \\
& \le C(m_1) \sup_{0 \le a < m_1} \left( \|f_{m_1, a, m_1-a}^{(1),V}\|_{L^2([-M_V, M_V]^c)}^2 + \|f_{m_1, a, m_1-a}^{(2),V}\|_{L^2([-M_V, M_V]^c)}^2 \right) \\
& \qquad \times \left( \|h\|_{L^2(\mu_V)}^2 + \|\mathcal{H}[\rho_V \Xi^{-1}[h]]\|_{L^2(\mu_V)}^2 \right) \\
& \le C(V, m_1) \|\rho_V\|_{L^\infty(\mathbb{R})} \left( \|h\|_{L^2(\mathbb{R})}^2 + \|\rho_V\|_{L^\infty(\mathbb{R})} \pi^2 \|\Xi^{-1}[h]\|_{L^2(\mu_V)}^2 \right) \\
& \le C(V, m_1) \left( \|h\|_{L^2(\mathbb{R})}^2 + \|\rho_V\|_{L^\infty(\mathbb{R})} \pi^2 C_L^2 \|h'\|_{L^2(\mu_V)}^2 \right) \\
& \le C_1(V, m_1) \|\partial_2^{m_2} \dots \partial_p^{m_p} f(\cdot, x_2, \dots, x_p)\|_{H^1(\mathbb{R})}^2
\end{aligned}
$$

where at the end, the constant $C_1(V, m_1)$ is defined by

$$
\begin{aligned}
C_1(V, m_1) := C(m_1) \max_{0 \le a < n_1} & \left( \|f_{n_1, a, n_1-a}^{(1),V}\|_{L^2([-M_V, M_V]^c)}^2 + \|f_{n_1, a, n_1-a}^{(2),V}\|_{L^2([-M_V, M_V]^c)}^2 \right) \\
& \times \|\rho_V\|_{L^\infty(\mathbb{R})} \left( 1 + \|\rho_V\|_{L^\infty(\mathbb{R})}^2 \pi^2 C_L^2 \right).
\end{aligned}
\quad (42)
$$
Above, the first integral that appears is well-defined, since by Lemma 4.13, one can check by assumption (v) that $f_{m_1,a,m_1-a}^{(1)}(x_1)$ and $f_{m_1,a,m_1-a}^{(2)}(x_1)$ behave like $O_{|x_1|\to\infty}(V'(x_1)^{-2})$ which is integrable by assumption (v) again.

In the following, we set $\mathfrak{g} := g - c$. We can now replace $g'$ by $\mathfrak{g}'$ since we handled all the terms involving $c$. By Cauchy-Schwarz inequality, with $C_2(V, m_1)$ defined by

$$C_2(V, n_1) = C(n_1) \max_{\substack{0 \le a < n_1 \\ 0 \le b \le n_1 - a}} \left( \left\| f_{n_1, a, b}^{(1), V} \right\|_{L^\infty([-M_V, M_V]^c)}^2 + \left\| f_{n_1, a, b}^{(3), V} \right\|_{L^2([-M_V, M_V]^c)}^2 + \left\| f_{n_1, a, b}^{(4), V} \right\|_{L^2([-M_V, M_V]^c)}^2 \right), \quad (43)$$

with

$$f_{n_1, a, b}^{(3), V} : x \mapsto \left. \frac{Q_a^{n_1} (\theta, \dots, \theta^{(a)}) (x)}{\rho_V(x)} \right|_{x \to \infty} \int_x^{\operatorname{sgn}(x)\infty} dt \rho_V(t)^2 \alpha(t)^2 P_b^{n_1-a} (\alpha, \dots, \alpha^{(b)}) (t)^2 \Big|_{t=\infty}^{1/2}, \quad (44)$$

$$f_{n_1, a, b}^{(4), V} : x \mapsto \left. \frac{Q_a^{n_1} (\theta, \dots, \theta^{(a)}) (x)}{\rho_V(x)} \right|_{x \to \infty} \int_x^{\operatorname{sgn}(x)\infty} dt \rho_V(t)^2 \left[ \alpha P_b^{n_1-a} (\alpha, \dots, \alpha^{(b)}) \right]' (t) \Big|_{t=\infty}^{1/2}, \quad (45)$$

we get

$$\sum_{a=0}^{m_1-1} \sum_{b=0}^{m_1-a} \int_{M_V}^{+\infty} dx_1 Q_a^{m_1} (\theta, \dots, \theta^{(a)}) (x_1)^2 \left\{ \begin{aligned} & \mathfrak{g}^{(m_1-a-b)}(x_1)^2 \alpha(x_1)^2 P_b^{m_1-a} (\alpha, \dots, \alpha^{(b)}) (x_1)^2 \\ & + \frac{1}{\rho_V(x_1)^2} \left[ \int_{x_1}^{\operatorname{sgn}(x_1)\infty} dt \rho_V(t) \mathfrak{g}^{(m_1-a-b+1)}(t) \alpha(t) P_b^{m_1-a} (\alpha, \dots, \alpha^{(b)}) (t) \right]^2 \right. \\ & \left. + \frac{1}{\rho_V(x_1)^2} \left[ \int_{x_1}^{\operatorname{sgn}(x_1)\infty} dt \rho_V(t) \mathfrak{g}^{(m_1-a-b)}(t) \left[ \alpha P_b^{m_1-a} (\alpha, \dots, \alpha^{(b)}) \right]' (t) \right]^2 \end{aligned} \right\} \\ & \le C_2(V, m_1) \|\mathfrak{g}\|_{H^{m_1+1}(\mathbb{R})}^2.$$

Finally, by using that $\mathfrak{g} = -h - 2PH[\rho_V \Xi^{-1}[h]]$, that $\pi^{-1}\mathcal{H}$ is an isometry in $L^2(\mathbb{R})$ and that for all $u \in H^1(\mathbb{R})$, $\mathcal{H}[u]' = \mathcal{H}[u']$, we obtain:

$$\left\| \Xi^{-1}[h]^{(m_1)} \right\|_{L^2([M_V, +\infty[)} \le 2 \max_{i=1,2} C_i(V, m_1)^{1/2} \left( \|h\|_{H^{m_1+1}(\mathbb{R})} + 2P\pi \|\rho_V \Xi^{-1}[h]\|_{H^{m_1+1}(\mathbb{R})} \right).$$

We now use the form stated in Lemma A.3, to conclude that

$$\left\| \Xi_1^{-1}[h]^{(m_1)} \right\|_{L^2([M_V, +\infty[)} \le 2 \max_{i=1,2} C_i(V, m_1)^{1/2} (2P\pi C_3(V, m_1) + 1) \|h\|_{H^{m_1+1}(\mathbb{R})}.$$

where $C_3(V, m_1)$ is explicitly given in Lemma A.3. By the exact same bounds, on $]-\infty, -M_V]$, we finally obtain

$$\left\| \Xi^{-1}[h]^{(m_1)} \right\|_{L^2([-M_V, M_V]^c)} \le 4 \max_{i=1,2} C_i(V, m_1)^{1/2} (2P\pi C_3(V, m_1) + 1) \|h\|_{H^{m_1+1}(\mathbb{R})}.$$
Now relaxing the dependance on $x_2, \dots, x_p \in \mathbb{R}$ and integrating with respect to these variables, we get

$$ \|\partial^{\underline{m}\Xi_1^{-1}}[f]\|_{L^2([-M_V, M_V]^c \times \mathbb{R}^{p-1})} \le 4 \max_{i=1,2} C_i(V, m_1)^{1/2} (2P\pi C_3(V, m_1) + 1) \\ \times \left( \int_{\mathbb{R}^{p-1}} \|\partial_2^{m_2} \dots \partial_p^{m_p} f(\cdot, x_2, \dots, x_p)\|_{H^{m_1+1}(\mathbb{R})}^2 dx_2 \dots dx_p \right)^{1/2}. $$

Thus we deduce that for a constant $C(n) > 0$ only depending on $n$ such that

$$ \|\Xi_1^{-1}[f]\|_{H^n([-M_V, M_V]^c \times \mathbb{R}^{p-1})} \le C(n) \max_{m_1 \le n} \max_{i=1,2} C_i(V, m_1)^{1/2} (2P\pi C_3(V, m_1) + 1) \|f\|_{H^{n+1}(\mathbb{R}^p)}. \quad (46) $$

Now, we prove the control on $[-M_V, M_V] \times \mathbb{R}^{p-1}$, we fix $x_2, \dots, x_p \in \mathbb{R}$. By Cauchy-Schwarz inequality:

$$ \int_{-M_V}^{M_V} |\Xi^{-1}[h]^{(m_1)}(x)|^2 dx \le \|\rho_V^{-1}\|_{L^\infty([-M_V, M_V])}^2 \|\rho_V \Xi^{-1}[h]^{(m_1)}\|_{L^2(\mathbb{R})}^2 \\ \le C_3(V, m_1)^2 \|\rho_V^{-1}\|_{L^\infty([-M_V, M_V])}^2 \|h\|_{H^{m_1}(\mathbb{R})}^2 $$

where the last inequality comes from Lemma A.3. Again relaxing the dependance on $x_2, \dots, x_p \in \mathbb{R}$ and integrating with respect to these variables, we get for a constant $C(n) > 0$ independent of $n$:

$$ \|\Xi_1^{-1}[f]\|_{H^n([-M_V, M_V] \times \mathbb{R}^{p-1})} \le C(n) \max_{m_1 \le n} C_3(V, m_1) \|\rho_V^{-1}\|_{L^\infty([-M_V, M_V])} \|f\|_{H^{n+1}(\mathbb{R}^p)}. $$

Collecting the last bound, using that $C_3(V, m)$ is increasing in $m$ and (46) leads to the conclusion for

$$ \mathcal{C}(\widetilde{\Xi}_1^{-1}, H^n) := C(n) \left[ C_3(V, n) \|\rho_V^{-1}\|_{L^\infty([-M_V, M_V])} + (2P\pi C_3(V, n) + 1) \max_{n_1 \le n} \max_{i=1,2} C_i(V, n_1)^{\frac{1}{2}} \right] \quad (47) $$

The fact that, upon choosing the potential $V_{\phi,t}$ with $\phi \in \cap_{k \ge 0} H^k(\mathbb{R})$, $t \mapsto \mathcal{C}(\widetilde{\Xi}_1^{-1}, H^n)$ is continuous is shown in Proposition B.8. $\square$

**Theorem 4.15** ($W_n^\infty(\mathbb{R})$-continuity of $\widetilde{\Xi}_1^{-1}$) Let $n \ge 1$, for all $f \in W_{n+1}^\infty(\mathbb{R}^p)$,

$$ \|\widetilde{\Xi}_1^{-1}[f]\|_{W_n^\infty(\mathbb{R}^p)} \le \mathcal{C}(\widetilde{\Xi}_1^{-1}, W_n^\infty) \|f\|_{W_{n+1}^\infty(\mathbb{R}^p)}. $$

Under the choice of potential $V_{\phi,t}$ defined in Theorem 1.3, for $\phi \in \cap_{k \ge 0} H^k(\mathbb{R})$ the map $t \in [0, 1] \mapsto \mathcal{C}(\widetilde{\Xi}_1^{-1}, W_n^\infty)$ is continuous.

**Proof** Let $f \in W_{n+1}(\mathbb{R}^p)$, let $m \le n$ and $(m_1, \dots, m_p) \in \mathbb{N}^p$ be such that $\sum_{i=1}^p m_i = m$, let $x_2, \dots, x_p \in \mathbb{R}^{p-1}$, we set $h: x_1 \mapsto \partial_2^{m_2} \dots \partial_p^{m_p} f(x_1, \dots, x_p)$ we know by theorem 4.11 that

$$ \widetilde{\Xi}^{-1}[h](x_1) = \frac{1}{\rho_V(x_1)} \int_{x_1}^{\operatorname{sgn}(x_1)\infty} dt \rho_V(t)g(t), $$
where $g := -\mathcal{X}[h] + 2P\mathcal{X} \circ \mathcal{H}[\rho_V \Xi^{-1}[h]]$. For the following we define

$$ \mathcal{I}_1^V : x \mapsto \frac{1}{\rho_V(x)} \left| \int_x^{\operatorname{sgn}(x)\infty} \rho_V(t) dt \right|, \qquad \mathcal{I}_2^V : x \mapsto \frac{1}{\rho_V(x)} \left| \int_x^{\operatorname{sgn}(x)\infty} \rho_V(t)^2 dt \right|^{1/2} . \quad (48) $$

By integration by parts, one can see that $\mathcal{I}_1^V(x) = \mathcal{O}_{|x|\to\infty}(V'(x)^{-1})$ is bounded on $\mathbb{R}$. So for the first and third term, by direct bounds:

$$ \left| \frac{1}{\rho_V(x_1)} \int_{x_1}^{\operatorname{sgn}(x_1)\infty} dt \rho_V(t) \left( -h(t) + \int_{\mathbb{R}} h(s) d\mu_V(s) \right) \right| \le 2 \|\mathcal{I}_1^V\|_{\infty} \|h\|_{\infty} \le 2 \|\mathcal{I}_1^V\|_{\infty} \|f\|_{W_{n+1}^{\infty}(\mathbb{R}^p)}. $$

For the two last terms, we want to use that $\pi^{-1}\mathcal{H}$ is an isometry on $L^2(\mathbb{R})$, so we use Cauchy-Schwarz inequality and the fact that $\mathcal{I}_2^V(x) = \mathcal{O}_{|x|\to\infty}(V'(x)^{-1/2})$ is bounded on $\mathbb{R}$ so that:

$$ \sup_{x_1 \in \mathbb{R}} \left| \frac{2P}{\rho_V(x_1)} \int_{x_1}^{\operatorname{sgn}(x_1)\infty} dt \rho_V(t) \mathcal{X} \circ \mathcal{H} [\rho_V \Xi^{-1}[h]](t) \right| \le C_4(V) \|h'\|_{\infty} \le C_4(V) \|f\|_{W_{n+1}^{\infty}(\mathbb{R}^p)}. $$

with

$$ C_4(V) := 2P \left( \|\mathcal{I}_2^V\|_{\infty} + \|\mathcal{I}_1^V\|_{\infty} \|\rho_V\|_{\infty}^{1/2} \right) \pi \|\rho_V\|_{\infty}^{1/2} C_{\mathcal{L}}. \quad (49) $$

Thus, by taking the supremum of $x_2, \dots, x_p \in \mathbb{R}^{p-1}$, we conclude that for $m_1 = 0$,

$$ \left\| \partial^{\underline{m}\Xi^{-1}} [f] \right\|_{L^{\infty}(\mathbb{R}^p)} \le (2 \|\mathcal{I}_1^V\|_{\infty} + C_4(V)) \|f\|_{W_{n+1}^{\infty}(\mathbb{R}^p)}. \quad (50) $$

For $m_1 \ne 0$ by (38), for all $|x| > M_V$, cf Lemma 4.9:

$$ \Xi^{-1}[h]^{(m_1)}(x_1) = \sum_{a=0}^{m_1-1} \sum_{b=0}^{m_1-a} \frac{Q_a^{m_1} (\theta, \dots, \theta^{(a)}) (x_1)}{\rho_V(x_1)} \int_{x_1}^{\operatorname{sgn}(x_1)\infty} g^{(m_1-a-b)}(t) P_b^{m_1-a}(\alpha, \dots, \alpha^{(b)})(t) d\mu_V(t) $$

where $g := -\mathcal{X}[h] - 2PW \circ \mathcal{L}^{-1}[h]$. Furthermore, setting

$$ c := \int_{\mathbb{R}} h(t) d\mu_V(t) - 2P \int_{\mathbb{R}} \mathcal{H} [\rho_V \Xi^{-1}[h]] (t) d\mu_V(t) $$

and

$$ f_{l,a,b}^{(5),V} : x \mapsto \frac{Q_a^l (\theta, \dots, \theta^{(a)}) (x)}{\rho_V(x)} \int_x^{\operatorname{sgn}(x)\infty} \left| P_b^{l-a} (\alpha, \dots, \alpha^{(b)})(t) \right| \rho_V(t) dt, \quad (51) $$

we can bound every term in the previous sum, involving $c$, namely taking $b = m_1 - a$:

$$ |c f_{m_1, a, m_1-a}^{(5)}(x_1)| \le (1 + 2P\pi \|\rho_V\|_{\infty} C_{\mathcal{L}}) \|h\|_{W_1^{\infty}(\mathbb{R})} \|f_{m_1, a, m_1-a}^{(5)}\|_{L^{\infty}([-M_V, M_V]^c)}. $$

We directly bound $c$ in the LHS, while to bound $f_{m_1, a, m_1-a}^{(5)}(x_1)$, we successfully applied Jensen's inequality, used the isometry property of $\pi^{-1}\mathcal{H}$ on $L^2(\mathbb{R})$ and used the inequality (26). Furthermore,
the fact that $f_{m_1,a,m_1-a}^{(5)}$ is bounded on $[-M_V, M_V]^c$ comes from Lemma 4.13. Finally, by setting $\mathfrak{g} := g - c$, it only remains to establish the following bounds:

$$
\begin{aligned}
& \left| \sum_{a=0}^{m_1-1} \sum_{b=0}^{m_1-a} Q_a^{m_1}(\theta, \dots, \theta^{(a)})(x_1) \frac{1}{\rho_V(x_1)} \int_{x_1}^{\operatorname{sgn}(x_1)\infty} \mathfrak{g}^{(m_1-a-b)}(t) P_b^{m_1-a}(\alpha, \dots, \alpha^{(b)})(t) \rho_V(t) dt \right| \\
& \le \sum_{a=0}^{m_1-1} \sum_{b=0}^{m_1-a} \left\{ \left\| \frac{Q_a^{m_1}(\theta, \dots, \theta^{(a)})}{\rho_V} \right\|_{\operatorname{sgn}(\cdot)\infty} \int_{\cdot}^{\operatorname{sgn}(\cdot)\infty} \left| \mathcal{H} \left[ (\rho_V \Xi^{-1}[h])^{(m_1-a-b)} \right] (t) \right| \cdot \left| P_b^{m_1-a}(\alpha, \dots, \alpha^{(b)})(t) \right| \rho_V(t) dt \right\|_{L^\infty([-M_V, M_V]^c)} \\
& \le \sum_{a=0}^{m_1-1} \sum_{b=0}^{m_1-a} \left\{ \begin{aligned}
& \|h\|_{W_{m_1}^\infty(\mathbb{R})} \|f_{m_1,a,b}^{(5),V}\|_{L^\infty([-M_V, M_V]^c)} \\
& + \|f_{m_1,a,b}^{(6),V}\|_{L^\infty([-M_V, M_V]^c)} \left\| \mathcal{H} \left[ (\rho_V \Xi^{-1}[h])^{(m_1-a-b)} \right] \right\|_{L^2(\mathbb{R})}
\end{aligned} \right\}
\end{aligned}
$$

with

$$ f_{l,a,b}^{(6),V} : x \mapsto \left| Q_a^l(\theta, \dots, \theta^{(a)})(x) \right| \frac{1}{\rho_V(x)} \sqrt{\int_{x}^{\operatorname{sgn}(x)\infty} \left| P_b^{l-a}(\alpha, \dots, \alpha^{(b)})(t) \right|^2 \rho_V(t)^2 dt}. \quad (52) $$

For each $a, b$, $f_{m_1,a,b}^{(6),V}$ is bounded on $[-M_V, M_V]^c$ because of Lemma 4.13 and Lemma A.2. By Cauchy-Schwarz inequality and Lemma A.3, we get:

$$ \left\| \mathcal{H} \left[ (\rho_V \Xi^{-1}[h])^{(m_1-a-b)} \right] \right\|_{L^2(\mathbb{R})} \le \pi \left\| (\rho_V \Xi^{-1}[h])^{(m_1-a-b)} \right\|_{L^2(\mathbb{R})} \le \pi C_6(V, m_1) \|h\|_{W_{m_1}^{\infty}(\mathbb{R})}. $$

Finally by the same reasonings as before, we get

$$
\begin{aligned}
& \left| \sum_{a=0}^{m_1-1} \sum_{b=0}^{m_1-a} \frac{Q_a^{m_1}(\theta, \dots, \theta^{(a)})(x_1)}{\rho_V(x_1)} \int_{x_1}^{\operatorname{sgn}(x_1)\infty} \mathfrak{g}^{(m_1-a-b)}(t) P_b^{m_1-a}(\alpha, \dots, \alpha^{(b)})(t) \rho_V(t) dt \right| \\
& \le C_7(V, m_1) \|h\|_{W_{m_1}^{\infty}(\mathbb{R})}
\end{aligned}
$$

where $C_7(V, m_1)$ is defined by

$$ C_7(V, m_1) := \max_{1 \le l \le m_1} l^2 \left( \max_{\substack{0 \le a < l \\ 0 \le b \le l-a}} \|f_{l,a,b}^{(5),V}\|_{L^\infty([-M_V, M_V]^c)} + \pi C_6(V, l) \max_{\substack{0 \le a < l \\ 0 \le b \le l-a}} \|f_{l,a,b}^{(6),V}\|_{L^\infty([-M_V, M_V]^c)} \right). \quad (53) $$

Thus, we deduce that

$$
\begin{aligned}
& \|\Xi^{-1}[h]^{(m_1)}\|_{L^\infty([-M_V, M_V]^c)} \\
& \le \left[ m_1(1 + 2P\pi \|\rho_V\|_\infty C_\mathcal{L}) \max_{0 \le a < m_1} \|f_{m_1, a, m_1-a}^{(5),V}\|_{L^\infty([-M_V, M_V]^c)} + C_7(V, m_1) \right] \|h\|_{W_{m_1}^\infty(\mathbb{R})}.
\end{aligned}
\quad (54)
$$
Now let $x \in [-M_V, M_V]$, by differentiating $m_1$ times (36), the Leibniz formula ensures that there exists polynomials $R_{m_1-a}^{m_1}$ depending on $(\theta, \dots, \theta^{(m_1-1-a)})$ and a polynomial $S^{m_1}$ of degree $m_1-1$ depending on $(\theta, \dots, \theta^{(m_1)})$, whose coefficients are independent of $V$ such that

$$ \widetilde{\Xi^{-1}[h]}^{(m_1)}(x) = \frac{S^{m_1}(\theta, \dots, \theta^{(m_1)})(x)}{\rho_V(x)} \int_x^{\operatorname{sgn}(x)\infty} g(t)\rho_V(t)dt + \sum_{a=0}^{m_1-2} R_{m_1-a}^{m_1}(\theta, \dots, \theta^{(m_1-1-a)})(x)g^{(a)}(x) - g^{(m_1-1)}(x). \quad (55) $$

We recall that the function $g$ is defined by

$$ g := -h + \int_{\mathbb{R}} h(t)d\mu_V(t) + 2P \left( \mathcal{H}[\rho_V \widetilde{\Xi^{-1}[h]}] - \int_{\mathbb{R}} \mathcal{H}[\rho_V \widetilde{\Xi^{-1}[h]}](t)d\mu_V(t) \right). $$

Then, for all, $x \in [-M_V, M_V]$, by the same bounds as before with $C_8(V, m_1)$ defined by

$$ C_8(V, m_1) := \max_{1 \le l \le m_1} \left( \|S^l(\theta, \dots, \theta^{(l)})\|_{L^\infty([-M_V, M_V])} \|\rho_V^{-1}\|_{L^\infty([-M_V, M_V])} \right. \\ \left. + l \max_{0 \le a \le l-2} \|R_{l-a}^l(\theta, \dots, \theta^{(l-1-a)})\|_{L^\infty([-M_V, M_V])} + 1 \right), \quad (56) $$

we obtain:

$$ \left| \frac{S^{m_1}(\theta, \dots, \theta^{(m_1)})(x_1)}{\rho_V(x_1)} \int_{x_1}^{\operatorname{sgn}(x_1)\infty} \mathcal{X}[h](t)\rho_V(t)dt + \sum_{a=0}^{m_1-2} R_{m_1-a}^{m_1}(\theta, \dots, \theta^{(m_1-1-a)})(x_1)\mathcal{X}[h]^{(a)}(x_1) - \mathcal{X}[h]^{(m_1-1)}(x_1) \right| \le 2C_8(V, m_1)\|h\|_{W_{m_1-1}^{\infty}(\mathbb{R})} $$

and

$$ 2P \left| \frac{S^{m_1}(\theta, \dots, \theta^{(m_1)})(x_1)}{\rho_V(x_1)} \int_{x_1}^{\operatorname{sgn}(x_1)\infty} \rho_V(t)dt + R_{m_1}^{m_1}(\theta, \dots, \theta^{(m_1-1)}) (x_1) \right| \int_{\mathbb{R}} |\mathcal{H}[\rho_V \widetilde{\Xi^{-1}[h]}](t)| d\mu_V(t) \le 2P\pi \|\rho_V\|_{\infty}^{1/2} C_L C_8(V, m_1) \|h\|_{W_1^{\infty}(\mathbb{R})}. $$

It remains to bound the terms involving the Hilbert-transform. For that, we use that for all $\phi \in H^{m_1}(\mathbb{R})$ and $a \in [0, m_1 - 1]$,

$$ |\mathcal{H}[\phi]^{(a)}(x)| = \sqrt{\mathcal{H}[\phi^{(a)}](x)^2} = \sqrt{\int_{+\infty}^{x} 2\mathcal{H}[\phi^{(a)}](t)\mathcal{H}[\phi^{(a+1)}](t)dt} \le \pi \sqrt{2\|\phi^{(a)}\|_{L^2(\mathbb{R})}\|\phi^{(a+1)}\|_{L^2(\mathbb{R})}}. $$

Applying those results for $\phi = \rho_V \widetilde{\Xi^{-1}[h]} \in H^{m_1}(\mathbb{R})$, Lemma 4.8 allows us to conclude that

$$ \|\mathcal{H}[\phi]\|_{W_{m_1-1}^{\infty}(\mathbb{R})} \le \sqrt{2\pi}\|\phi\|_{H^{m_1}(\mathbb{R})}. $$

We conclude by Lemma A.3 that

$$ \|\mathcal{H}[\phi]\|_{W_{m_1-1}^{\infty}(\mathbb{R})} \le \sqrt{2\pi C_6(V, m_1)} \|h\|_{W_{m_1}^{\infty}(\mathbb{R})} $$
and thus, with $C_8(V, m_1)$ defined in (56) we get

$$2P \left| \frac{S^{m_1}(\theta, \dots, \theta^{(m_1)})(x_1)}{\rho_V(x_1)} \int_{x_1}^{\operatorname{sgn}(x_1)\infty} \mathcal{H}[\rho_V \Xi^{-1}[h]](t) \rho_V(t) dt + \sum_{a=0}^{m_1-2} R_{m_1-a}^{m_1}(\theta, \dots, \theta^{(m_1-1-a)})(x_1) \mathcal{H}[\rho_V \Xi^{-1}[h]]^{(a)}(x_1) - \mathcal{H}[\rho_V \Xi^{-1}[h]]^{(m_1-1)}(x_1) \right| \le 2P\sqrt{2\pi}C_6(V, m_1)C_8(V, m_1) \|h\|_{W_{m_1-1}^{\infty}(\mathbb{R})}.$$

All the previous bounds yield

$$\|\widetilde{\Xi^{-1}[h]}^{(m_1)}\|_{L^{\infty}([-M_V, M_V])} \le (2 + 2P\pi \|\rho_V\|_{\infty}^{1/2}C_{\mathcal{L}} + 2P\sqrt{2\pi}C_6(V, m_1)) C_8(V, m_1) \|h\|_{W_{m_1}^{\infty}(\mathbb{R})}.$$

Upon taking the supremum over $x_2, \dots, x_p \in \mathbb{R}^{p-1}$ in (12), and over $m \le n$ we conclude that

$$\|\widetilde{\Xi_1^{-1}[f]}\|_{W_n^{\infty}(\mathbb{R}^p)} \le \mathcal{C}(\widetilde{\Xi_1^{-1}}, W_n^{\infty}) \|f\|_{W_{n+1}^{\infty}(\mathbb{R}^p)}.$$

with $\mathcal{C}(\widetilde{\Xi_1^{-1}}, W_n^{\infty})$ defined as

$$\mathcal{C}(\widetilde{\Xi_1^{-1}}, W_n^{\infty}) = \max_{m_1 \le n} \left[ m_1(1 + 2P\pi \|\rho_V\|_{\infty} C_{\mathcal{L}}) \max_{0 \le a < l} \|f^{(5),V}_{m_1, a, m_1-a}\|_{L^{\infty}([-M_V, M_V]^c)} + C_7(V, m_1) \right] \\ + \max_{m_1 \le n} (2 + 2P\pi \|\rho_V\|_{\infty}^{1/2} C_{\mathcal{L}} + 2P\sqrt{2\pi}C_6(V, m_1)) C_8(V, m_1) + 2\|\mathcal{I}_1^V\|_{\infty} + C_4(V). \quad (57)$$

The fact that, upon choosing the potential $V_{\phi,t}$ with $\phi \in \cap_{k \ge 0} H^k(\mathbb{R})$, $t \mapsto \mathcal{C}(\widetilde{\Xi_1^{-1}}, W_n^{\infty})$ is continuous is shown in Proposition B.8. $\square$

Finally, we define the variable insertion operators which will also be involved in the loop equations.

**Definition 4.16** If $\phi$ is a function in $n$ variables, we define the $n$-th variable insertion operator $\Theta^{(p)}$ as

$$\Theta^{(p)}[\phi](\xi_1, \dots, \xi_{n-1}) = \phi(\xi_1, \dots, \xi_{p-1}, \xi_1, \xi_p, \dots, \xi_{n-1}) \quad (58)$$

**Corollary 4.17** Let $n, p \ge 1$, $a \in [2, p+1]$, for all $f \in W_{n+1}^{\infty}(\mathbb{R}^{p+1})$,

$$\|\Theta^{(a)} \circ \widetilde{\Xi_1^{-1}}[f]\|_{W_n^{\infty}(\mathbb{R}^p)} \le 2\mathcal{C}(\widetilde{\Xi_1^{-1}}, W_n^{\infty}) \|f\|_{W_{n+1}^{\infty}(\mathbb{R}^{p+1})}$$

where the constant $\mathcal{C}(\widetilde{\Xi_1^{-1}}, W_n^{\infty})$ was introduced in Theorem 4.15.

**Proof** Let $f \in W_{n+1}^{\infty}(\mathbb{R}^{p+1})$, $x_1, \dots, x_p \in \mathbb{R}$, $\underline{m} := (m_1, \dots, m_p) \in \mathbb{N}^p$ such that $m := \sum_{i=1}^p m_i \le n$.

$$\partial^{\underline{m}} \Theta^{(a)} \circ \widetilde{\Xi_1^{-1}}[f](x_1, \dots, x_p) = \partial_1^{m_1} \partial_2^{m_2} \dots \partial_{a-1}^{m_{a-1}} \partial_{a+1}^{m_a} \dots \partial_{p+1}^{m_p} \widetilde{\Xi_1^{-1}}[f](x_1, \dots, x_{a-1}, x_1, x_a, \dots, x_p) \\ + \partial_2^{m_2} \dots \partial_{a-1}^{m_{a-1}} \partial_a^{m_1} \partial_{a+1}^{m_a} \partial_{p+1}^{m_p} \widetilde{\Xi_1^{-1}}[f](x_1, \dots, x_{a-1}, x_1, x_a, \dots, x_p).$$

$$\text{Thus } \left\| \partial^{\underline{m}} \Theta^{(a)} \circ \widetilde{\Xi_1^{-1}} [f] \right\|_{L^{\infty}(\mathbb{R}^p)} \le 2 \left\| \widetilde{\Xi_1^{-1}} [f] \right\|_{W_n^{\infty}(\mathbb{R}^{p+1})} \le 2\mathcal{C}(\widetilde{\Xi_1^{-1}}, W_n^{\infty}) \|f\|_{W_{n+1}^{\infty}(\mathbb{R}^{p+1})}. \quad \square$$
The last control that we need is on $\Theta^{(a)} \circ \tilde{\Xi}_1^{-1}$ in $H^n$-norm. $\Theta^{(a)}$ was defined in Definition 4.16.

**Theorem 4.18 (H$^n$-continuity for $\Theta^{(a)} \circ \tilde{\Xi}_1^{-1}$)** Let $n \in \mathbb{N}$, $p \in [1, +\infty[$, let $a \in [2, p+1]$, there exists a constant $C(\Theta^{(a)} \circ \tilde{\Xi}_1^{-1}, H^n) > 0$, such that for all $f \in H^{n+1}(\mathbb{R}^{p+1})$,

$$ \|\Theta^{(a)} \circ \tilde{\Xi}_1^{-1}[f]\|_{H^n(\mathbb{R}^p)} \le C(\Theta^{(a)} \circ \tilde{\Xi}_1^{-1}, H^n)\|f\|_{H^{n+1}(\mathbb{R}^{p+1})}. $$

Under the choice of potential $V_{\phi,t}$ defined in Theorem 1.3, for $\phi \in \cap_{k \ge 0} H^k(\mathbb{R})$ the map $t \in [0, 1] \mapsto C(\Theta^{(a)} \circ \tilde{\Xi}_1^{-1}, H^n)$ is continuous.

**Proof** Let $f \in H^{n+1}(\mathbb{R}^{p+1})$, Let $f \in W_{n+1}^{\infty}(\mathbb{R}^{p+1})$, $x_1, \dots, x_p \in \mathbb{R}$, $\underline{m} := (m_1, \dots, m_p) \in \mathbb{N}^p$ such that $m := \sum_{i=1}^p m_i \le n$. We set $h : (x, y) \mapsto \partial_2^{m_2} \dots \partial_{a-1}^{m_{a-1}} \partial_{a+1}^{m_a} \dots \partial_{p+1}^{m_p} f(x, x_2, \dots, x_{a-1}, y, x_a, \dots, x_p)$ and

$$ g(x, y) := -h(x, y) + 2PH[\rho_V \Xi^{-1}[h(., y)]](x) + c(y) $$

where we have set $c(y) = \int_{\mathbb{R}} h(s, y)d\mu_V(s) - 2P \int_{\mathbb{R}} \mathcal{H}[\rho_V \Xi^{-1}[h(., y)]](s)d\mu_V(s)$. Let $x \in \mathbb{R}$,

$$ \partial^{\underline{m}} \Theta^{(a)} \circ \tilde{\Xi}_1^{-1}[f](x_1, \dots, x_p) = \Xi^{-1}[\partial_2^{m_1} h(., x_1)](x_1) + \Xi^{-1}[h(., x_1)]^{(m_1)}(x_1). \quad (59) $$

The first term is easy to control by Theorem 4.11,

$$ \widetilde{\Xi}^{-1}[\partial_a^{m_1} h(., x_1)](x_1) = \frac{1}{\rho_V(x_1)} \int_{x_1}^{\operatorname{sgn}(x_1)\infty} dt \rho_V(t) \left\{ \begin{aligned} & -\partial_2^{m_1} h(t, x_1) + \int_{\mathbb{R}} \partial_2^{m_1} h(s, x_1) d\mu_V(s) \\ & + 2PH[\rho_V \widetilde{\Xi}^{-1}[\partial_a^{m_1} h(., x_1)]](t) - 2P \int_{\mathbb{R}} 2PH[\rho_V \widetilde{\Xi}^{-1}[\partial_a^{m_1} h(., x_1)]](s) d\mu_V(s) \end{aligned} \right\}. $$

From the standard arguments that we used before,

$$ \int_{\mathbb{R}} \widetilde{\Xi}^{-1}[\partial_2^{m_1} h(., x_1)](x_1)^2 dx_1 \le 4(1 + 4P^2\pi^2 \|\rho_V\|_{L^{\infty}(\mathbb{R})}^2 C_L^2) \left\{ \begin{aligned} & \|\mathcal{I}_2^V\|_{L^{\infty}(\mathbb{R})}^2 \|h\|_{H^{m_1+1}(\mathbb{R}^2)}^2 \\ & + \|\rho_V\|_{L^{\infty}(\mathbb{R})} \|\mathcal{I}_1^V\|_{L^{\infty}(\mathbb{R})}^2 \|h\|_{H^{m_1+1}(\mathbb{R}^2)}^2 \end{aligned} \right\} \quad (60) $$

where $\mathcal{I}_1^V$ and $\mathcal{I}_2^V$ have been defined in (48).

We now deal with the second term in (59). By (39), we have for a constant $C(m_1) > 0$ depending only on $m_1$:
$$
\int_{M_V}^{+\infty} \Xi^{-1}[h(., x_1)]^{(m_1)}(x_1)^2 dx_1 \le C(m_1) \sum_{a=0}^{m_1-1} \sum_{b=0}^{m_1-a} \int_{M_V}^{+\infty} dx_1 Q_a^{m_1} (\theta, \dots, \theta^{(a)})(x_1)^2 \\
\times \left\{ [\partial_1^{m_1-a-b} g(x_1, x_1)]^2 \alpha(x_1)^2 P_b^{m_1-a}(\alpha, \dots, \alpha^{(b)})(x_1)^2 \right. \\
+ \frac{1}{\rho_V(x_1)^2} \left[ \int_{x_1}^{\operatorname{sgn}(x_1)\infty} dt \rho_V(t) \partial_1^{m_1-a-b+1} g(t, x_1) \alpha(t) P_b^{m_1-a}(\alpha, \dots, \alpha^{(b)})(t) \right]^2 \\
\left. + \frac{1}{\rho_V(x_1)^2} \left[ \int_{x_1}^{\operatorname{sgn}(x_1)\infty} dt \rho_V(t) \partial_1^{m_1-a-b} g(t, x_1) [\alpha P_b^{m_1-a}(\alpha, \dots, \alpha^{(b)})]'(t) \right]^2 \right\} \cdot x \quad (61)
$$

We first deal with the presence of c in the sum. This term only arises in the sum when $b = m_1 - a$.
By using the functions $f_{m_1,a,b}^{(j),V}$ defined in (40), we can bound these terms, for all $a \in [0, m_1 - 1]$, by

$$
\int_{M_V}^{+\infty} dx_1 \mathfrak{c}(x_1)^2 \left( \mathfrak{f}_{m_1,a,m_1-a}^{(1)}(x_1)^2 + \mathfrak{f}_{m_1,a,m_1-a}^{(2),V}(x_1)^2 \right) \le 2 \left( \sum_{j \in \{1,2\}} \| \mathfrak{f}_{m_1,a,m_1-a}^{(j),V} \|_{L^\infty([-M_V,M_V]^c)}^2 \right) \\
\times \int_{\mathbb{R}} dx_1 \left( \int_{\mathbb{R}} h(t,x_1)^2 d\mu_V(t) + 4P^2 \int_{\mathbb{R}} \mathcal{H} \left[ \rho_V \Xi^{-1} [h(.,x_1)] \right] (t)^2 d\mu_V(t) \right) \\
\le 2 \left( \sum_{j \in \{1,2\}} \| \mathfrak{f}_{m_1,a,m_1-a}^{(j),V} \|_{L^\infty([-M_V,M_V]^c)}^2 \right) \left( \| \rho_V \|_{L^\infty(\mathbb{R})} + 4P^2 \pi^2 \| \rho_V \|_{L^\infty(\mathbb{R})}^3 C_L^2 \right) \| h \|_{H^1(\mathbb{R}^2)}^2.
$$

Since we handled all the terms involving $c$, it just remains to bound (61) with the substitution $g(x, y) \rightsquigarrow g(x, y) := g(x, y) - c(y)$, namely:

$$
\int_{M_V}^{+\infty} dx_1 Q_a^{m_1} (\theta, \dots, \theta^{(a)}) (x_1)^2 \left\{ \partial_1^{m_1-a-b} \mathfrak{g}(x_1, x_1)^2 \alpha(x_1)^2 P_b^{m_1-a} (\alpha, \dots, \alpha^{(b)})(x_1)^2 \right. \\
+ \frac{1}{\rho_V(x_1)^2} \left[ \int_{x_1}^{\operatorname{sgn}(x_1)\infty} dt \rho_V(t) \partial_1^{m_1-a-b+1} \mathfrak{g}(t, x_1) \alpha(t) P_b^{m_1-a} (\alpha, \dots, \alpha^{(b)})(t) \right]^2 \\
\left. + \frac{1}{\rho_V(x_1)^2} \left[ \int_{x_1}^{\operatorname{sgn}(x_1)\infty} dt \rho_V(t) \partial_1^{m_1-a-b} \mathfrak{g}(t, x_1) [\alpha P_b^{m_1-a} (\alpha, \dots, \alpha^{(b)})]'(t) \right]^2 \right\}.
$$

For the first term, we use the fact that for fixed $x \in \mathbb{R}$, $t \mapsto \partial_1^{m_1-a-b} g(t, x)$ goes to zero at infinity as an element of $H^1(\mathbb{R})$ and by Cauchy-Schwarz inequality, we get:

$$
|\partial_1^{m_1-a-b} \mathfrak{g}(x_1, x_1)| = \sqrt{\int_{+\infty}^{x_1} 2 \partial_1^{m_1-a-b} \mathfrak{g}(t, x_1) \partial_1^{m_1-a-b+1} \mathfrak{g}(t, x_1) dt} \le \sqrt{2} \|\mathfrak{g}(., x_1)\|_{H^{m_1-a-b+1}(\mathbb{R})}.
$$
Furthermore, for all $x_1 > M_V$, $f_{m_1,a,b}^{(1),V} : x \mapsto Q_a^{m_1}(\theta, \dots, \theta^{(a)})(x)\alpha(x)P_b^{m_1-a}(\alpha, \dots, \alpha^{(b)})(x)$ is bounded since it is continuous and a $O(|x|^{-\infty}(V'(x)^{-2})$ by Lemma 4.13. We conclude, by Lemma A.3 that, with $f_{m_1,a,b}^{(1),V}$ being given in (40),

$$
\begin{aligned}
& \sum_{a=0}^{m_1-1} \sum_{b=0}^{m_1-a} \int_{M_V}^{+\infty} dx_1^2 \partial_1^{m_1-a-b} \mathfrak{g}(x_1, x_1)^2 \mathfrak{f}_{m_1,a,b}^{(1),V}(x_1)^2 \\
& \le C(m_1) \max_{\substack{0 \le a < m_1 \\ 0 \le b \le m_1 - a}} \|\mathfrak{f}_{m_1,a,b}^{(1),V}\|_{L^\infty([-M_V, M_V]^c)}^2 \int_{M_V}^{+\infty} \|\mathfrak{g}(\cdot, x_1)\|_{H^{m_1-a-b+1}(\mathbb{R})}^2 dx_1 \\
& \le C(m_1) \max_{\substack{0 \le a < m_1 \\ 0 \le b \le m_1 - a}} \|\mathfrak{f}_{m_1,a,b}^{(1),V}\|_{L^\infty([-M_V, M_V]^c)}^2 \|\mathfrak{g}\|_{H^{m_1+1}(\mathbb{R}^2)}^2 \\
& \le C(m_1) [1 + 4P^2\pi^2 \|\rho_V\|_{L^\infty(\mathbb{R})} C_3(V, n)] \max_{\substack{0 \le a < m_1 \\ 0 \le b \le m_1 - a}} \|\mathfrak{f}_{m_1,a,b}^{(1),V}\|_{L^\infty([-M_V, M_V]^c)}^2 \|\partial_3^{m_3} f(\cdot, z)\|_{H^{m_1+1}(\mathbb{R}^2)}^2.
\end{aligned}
$$

It just remains to bound

$$
\int_{M_V}^{+\infty} dx_1 \frac{Q_a^{m_1}(\theta, \dots, \theta^{(a)})(x_1)^2}{\rho_V(x_1)^2} \left\{ \left[ \int_{x_1}^{\operatorname{sgn}(x_1)\infty} dt \rho_V(t) \partial_1^{m_1-a-b+1} \mathfrak{g}(t, x_1) \alpha(t) P_b^{m_1-a}(\alpha, \dots, \alpha^{(b)})(t) \right]^2 \right. \\
\left. + \left[ \int_{x_1}^{\operatorname{sgn}(x_1)\infty} dt \rho_V(t) \partial_1^{m_1-a-b} \mathfrak{g}(t, x_1) \left[ \alpha P_b^{m_1-a}(\alpha, \dots, \alpha^{(b)}) \right]'(t) \right]^2 \right\}.
$$

For the first term, we use Cauchy-Schwarz inequality, Lemma 4.13 and the function $f_{m_1,a,b}^{(3),V}$ defined in (44) to get

$$
\begin{aligned}
& \frac{|Q_a^{m_1}(\theta, \dots, \theta^{(a)})(x_1)|^2}{\rho_V(x_1)^2} \left| \int_{x_1}^{\operatorname{sgn}(x_1)\infty} dt \rho_V(t) \partial_1^{m_1-a-b+1} \mathfrak{g}(t, x_1) \alpha(t) P_b^{m_1-a}(\alpha, \dots, \alpha^{(b)})(t) \right|^2 \\
& \le \|f_{m_1,a,b}^{(3),V}\|_{L^\infty([-M_V, M_V]^c)}^2 \|\mathfrak{g}(\cdot, x_1)\|_{H^{m_1+1}(\mathbb{R})}^2 \\
& \le 2 \|f_{m_1,a,b}^{(3),V}\|_{L^\infty([-M_V, M_V]^c)}^2 (1 + 4P^2\pi^2 \|\rho_V\|_{L^\infty(\mathbb{R})} C_3(V, n)) \\
& \quad \times \|\partial_2^{m_2} \dots \partial_{a-1}^{m_{a-1}} \partial_{a+1}^{m_a} \dots \partial_{p+1}^{m_p} f(\cdot, x_2, \dots, x_{a-1}, x_1, x_a, \dots, x_p)\|_{H^{m_1+1}(\mathbb{R})}^2.
\end{aligned}
$$

We proceed in the exact same way for the second term and do the same thing on $]-\infty, -M_V]$, for every term we dealt with. Finally, by integrating with respect to $x_1$, collecting all the terms and then integrating over $x_1 \in [-M_V, M_V]^c$ and over $x_2, \dots, x_p \in \mathbb{R}$, we get

$$
\| \Theta^{(a)} [ \Xi^{-1} [ \partial_2^{m_2} \dots \partial_{a-1}^{m_{a-1}} \partial_{a+1}^{m_a} \dots \partial_{p+1}^{m_p} f ]^{(m_1)} ] \|_{L^2([-M_V, M_V]^c \times \mathbb{R}^{p-1})} \le C_{10}(V, n) \|f\|_{H^{n+1}(\mathbb{R}^{p+1})}
$$
with $C_{10}(V, n)$ defined by

$$ C_{10}(V, n)^2 := C(n) \max_{m_1 \le n} \left\{ \begin{aligned} & \left( \|\rho_V\|_{L^\infty(\mathbb{R})} + 4P^2\pi^2 \|\rho_V\|_{L^\infty(\mathbb{R})}^3 C_L^2 \right) \max_{\substack{0 \le a < m_1 \\ j \in \{1,2\}}} \|\mathfrak{f}_{m_1, a, m_1-a}^{(j),V}\|_{L^\infty([-M_V, M_V]^c)}^2 \\ & + [1 + 4P^2\pi^2 \|\rho_V\|_{L^\infty(\mathbb{R})} C_3(V, n)] \max_{\substack{0 \le a < m_1 \\ 0 \le b \le m_1-a \\ j \in \{1,3\}}} \|\mathfrak{f}_{m_1, a, b}^{(j),V}\|_{L^\infty([-M_V, M_V]^c)}^2 \end{aligned} \right\}. \quad (62) $$

It just remains to bound $\int_{-M_V}^{M_V} \widetilde{\Xi}^{-1}[h(., x_1)]^{(m_1)}(x_1)^2 dx_1$. For that, we use (55)

$$ \int_{-M_V}^{M_V} \widetilde{\Xi}^{-1}[h(., x_1)]^{(m_1)}(x_1)^2 dx_1 \\ \le C(m_1) \int_{-M_V}^{M_V} dx_1 \left\{ \frac{S^{m_1}(\theta, \dots, \theta^{(m_1)})(x_1)^2}{\rho_V(x_1)^2} \left( \int_{x_1}^{\operatorname{sgn}(x_1)\infty} g(t, x_1) \rho_V(t) dt \right)^2 \right. \\ \left. + \sum_{a=0}^{m_1-2} R_{m_1-a}^{m_1} (\theta, \dots, \theta^{(m_1-1-a)})(x_1)^2 [\partial_1^a g(x_1, x_1)]^2 + [\partial_1^{m_1-1} g(x_1, x_1)]^2 \right\} $$

By the same procedure as before, we first deal with $c$ defined at the beginning of the proof, this yields, with $\mathcal{I}_a$ defined in Theorem 4.15

$$ \int_{-M_V}^{M_V} c(x_1)^2 dx_1 \left[ \frac{S^{m_1}(\theta, \dots, \theta^{(m_1)})(x_1)^2}{\rho_V(x_1)^2} \left( \int_{x_1}^{\operatorname{sgn}(x_1)\infty} \rho_V(t) dt \right)^2 + R_{m_1}^{m_1} (\theta, \dots, \theta^{(m_1-1-a)})(x_1)^2 \right] \\ \le C(m_1) \|h\|_{H^1(\mathbb{R}^2)}^2 \|\rho_V\|_{L^\infty(\mathbb{R})} \left( 1 + 4P^2\pi^2 \|\rho_V\|_{L^\infty(\mathbb{R})}^2 C_L^2 \right) \left\{ \|\mathcal{I}_1^V\|_{\infty}^2 \|S^{m_1}(\theta, \dots, \theta^{(m_1)})\|_{L^\infty([-M_V, M_V])}^2 \right. \\ \left. + \|R_{m_1}^{m_1} (\theta, \dots, \theta^{(m_1-1)})\|_{L^\infty([-M_V, M_V])}^2 \right\}. $$

Hence as before, we can replace $g$ by $\mathfrak{g}$ and conclude with the last bounds:

$$ \int_{-M_V}^{M_V} \frac{S^{m_1}(\theta, \dots, \theta^{(m_1)})(x_1)^2}{\rho_V(x_1)^2} \left( \int_{x_1}^{\operatorname{sgn}(x_1)\infty} \mathfrak{g}(t, x_1) \rho_V(t) dt \right)^2 dx_1 \\ \le \|\mathfrak{g}\|_{L^2(\mathbb{R}^2)}^2 \|\mathcal{I}_2^V S^{m_1}(\theta, \dots, \theta^{(m_1)})\|_{L^\infty([-M_V, M_V])}^2 $$
and by Cauchy-Schwarz inequality,

$$
\begin{aligned}
& \int_{-M_V}^{M_V} dx_1 \left( \sum_{a=0}^{m_1-2} R_{m_1-a}^{m_1} (\theta, \dots, \theta^{(m_1-1-a)}) (x_1)^2 \partial_1^a \mathfrak{g}(x_1, x_1)^2 + \partial_1^{m_1-1} \mathfrak{g}(x_1, x_1)^2 \right) \\
& \le \left\{ 1 + \max_{a \in [0, m_1-2]} \left\| R_{m_1-a}^{m_1} (\theta, \dots, \theta^{(m_1-1-a)}) \right\|_{L^\infty([-M_V, M_V])}^2 \right\} \\
& \quad \times \sum_{a=0}^{m_1-1} \int_{-M_V}^{M_V} dx_1 \int_{+\infty}^{x_1} dt \partial_1^{a+1} \mathfrak{g}(t, x_1) \partial_1^a \mathfrak{g}(t, x_1) \\
& \le C(m_1) \left\{ 1 + \max_{a \in [0, m_1-2]} \left\| R_{m_1-a}^{m_1} (\theta, \dots, \theta^{(m_1-1-a)}) \right\|_{L^\infty([-M_V, M_V])}^2 \right\} \|\mathfrak{g}\|_{H^{m_1}(\mathbb{R}^2)}^2.
\end{aligned}
$$

Moreover, by Lemma A.3

$$
\|\mathfrak{g}\|_{H^{m_1}(\mathbb{R}^2)} \le \|h\|_{H^{m_1}(\mathbb{R}^2)} + 2P\pi \left\| \rho_V \Xi_1^{-1}[h] \right\|_{H^{m_1}(\mathbb{R}^2)} \le [1 + 2P\pi C_3(V, n)] \|h\|_{H^{m_1}(\mathbb{R}^2)}.
$$

We can then conclude that

$$
\begin{aligned}
& \int_{-M_V}^{M_V} dx_1 \left( \sum_{a=0}^{m_1-2} R_{m_1-a}^{m_1} (\theta, \dots, \theta^{(m_1-1-a)}) (x_1)^2 \partial_1^a \mathfrak{g}(x_1, x_1)^2 + \partial_1^{m_1-1} \mathfrak{g}(x_1, x_1)^2 \right) \\
& \le C(m_1) [1 + 2P\pi C_3(V, n)] \left\{ 1 + \max_{a \in [0, m_1-2]} \left\| R_{m_1-a}^{m_1} (\theta, \dots, \theta^{(m_1-1-a)}) \right\|_{L^\infty([-M_V, M_V])}^2 \right\} \|h\|_{H^{m_1}(\mathbb{R}^2)}^2.
\end{aligned}
$$

Thus by integrating with respect to $z \in \mathbb{R}$, we get,

$$
\left\| \Theta^{(a)} \left[ \Xi^{-1} \left[ \partial_2^{m_2} \dots \partial_{a-1}^{m_{a-1}} \partial_{a+1}^{m_a} \dots \partial_{p+1}^{m_p} f \right]^{(m_1)} \right] \right\|_{L^2([-M_V, M_V] \times \mathbb{R}^{p-1})} \le C_{11}(V, n) \|\partial_3^{m_3} f\|_{H^{m_1+1}(\mathbb{R}^{p+1})}
$$

with $C_{11}(V, n) > 0$ defined by

$$
\begin{aligned}
C_{11}(V, n)^2 := \max_{m_1 \le n} C(m_1) & \left\{ \|\rho_V\|_{L^\infty(\mathbb{R})} (1 + 4P^2\pi^2 \|\rho_V\|_{L^\infty(\mathbb{R})}^2 C_L^2) \right. \\
& \times \left\{ \|\mathcal{I}_1^V\|_{\infty}^2 \|S^{m_1}(\theta, \dots, \theta^{(m_1)})\|_{L^\infty([-M_V, M_V])}^2 + \|R_{m_1}^{m_1}(\theta, \dots, \theta^{(m_1-1)})\|_{L^\infty([-M_V, M_V])}^2 \right. \\
& \qquad \left. + [1 + 2P\pi C_3(V, n)]^2 \left( \|\mathcal{I}_2^V\|_{\infty}^2 \|S^{m_1}(\theta, \dots, \theta^{(m_1)})\|_{L^\infty([-M_V, M_V])}^2 \right. \right. \\
& \qquad \qquad \left. \left. + 1 + \max_{a \in [0, m_1-2]} \|R_{m_1-a}^{m_1}(\theta, \dots, \theta^{(m_1-1-a)})\|_{L^\infty([-M_V, M_V])}^2 \right) \right\}.
\end{aligned}
\quad (63)
$$

Collecting the bounds on the $L^2$-norms of $\Theta^{(a)}[\Xi^{-1}[\partial_2^{m_2} \dots \partial_{a-1}^{m_{a-1}} \partial_{a+1}^{m_a} \dots \partial_{p+1}^{m_p} f]^{(m_1)}]$ on $[-M_V, M_V]^c \times \mathbb{R}$ and $[-M_V, M_V] \times \mathbb{R}$, we obtain:

$$
\left\| \Theta^{(a)} \left[ \Xi^{-1} \left[ \partial_2^{m_2} \dots \partial_{a-1}^{m_{a-1}} \partial_{a+1}^{m_a} \dots \partial_{p+1}^{m_p} f \right]^{(m_1)} \right] \right\|_{L^2(\mathbb{R}^p)} \le 2 \max_{i \in \{10, 11\}} C_i(V, n) \|h\|_{H^{m_1+1}(\mathbb{R}^{p+1})}.
$$
By combining the above equation together with (60) and taking the supremum over $m \le n$, we get

$$ \|\Theta^{(a)} \circ \widetilde{\Xi}_1^{-1}[f]\|_{H^{n}(\mathbb{R}^p)} \le c(\Theta^{(a)} \circ \widetilde{\Xi}_1^{-1}, H^n) \|f\|_{H^{n+1}(\mathbb{R}^{p+1})} $$

with, $C_{10}$ and $C_{11}$ being given in (62) and (63), $\mathcal{I}_a^V$ being given in (48);

$$ \mathcal{C}(\Theta^{(a)} \circ \widetilde{\Xi}_1^{-1}, H^n) := 2 \max_{i \in \{10, 11\}} C_i(V, n) + 2 \sqrt{ (1 + 4P^2\pi^2 \|\rho_V\|_{L^\infty(\mathbb{R})}^2 C_\mathcal{I}_a^2) (\|\mathcal{I}_2^V\|_{L^\infty(\mathbb{R})}^2 + \|\rho_V\|_{L^\infty(\mathbb{R})} \|\mathcal{I}_1^V\|_{L^\infty(\mathbb{R})}^2) } . \quad (64) $$

This yields the conclusion. The fact that, upon choosing the potential $V_{\phi,t}$ with $\phi \in \cap_{k \ge 0} H^k(\mathbb{R})$, $t \mapsto \mathcal{C}(\Theta^{(a)} \circ \widetilde{\Xi}_1^{-1}, H^n)$ is continuous is shown in Proposition B.8. $\square$

# 5 Asymptotic expansion of the linear statistics

## 5.1 Loop equations for general functions

We are now ready to state the loop equations, we recall the definition of a linear statistic was defined in (3).

**Theorem 5.1 (Dyson-Schwinger equations)** The level 1 Schwinger-Dyson equation holds for all $\psi_1 \in \cap_{k \ge 0} H^k(\mathbb{R})$ and takes the form:

$$ \langle \psi_1 \rangle_{\mathcal{L}_N} = \frac{P}{N} \langle \partial_1 \Xi^{-1}[\psi_1] \rangle_{\mu_V} + \frac{P}{N} \langle \partial_1 \Xi^{-1}[\psi_1] \rangle_{\mathcal{L}_N} - P \langle \mathcal{D} \circ \Xi^{-1}[\psi_1] \rangle_{\mathcal{L}_N \otimes \mathcal{L}_N} . \quad (65) $$

For all $\psi_n \in \cap_{k \ge 0} H^k(\mathbb{R}^n)$, the level $n > 1$ Schwinger-Dyson equations reads:

$$ \langle \psi_n \rangle_{\bigotimes \mathcal{L}_N}^n = \frac{P}{N} \langle \partial_1 \widetilde{\Xi}_1^{-1}[\psi_n] \rangle_{\mu_V \bigotimes \mathcal{L}_N}^{n-1} + \frac{P}{N} \langle \partial_1 \widetilde{\Xi}_1^{-1}[\psi_n] \rangle_{\bigotimes \mathcal{L}_N}^n - P \langle \mathcal{D}_1 \circ \widetilde{\Xi}_1^{-1}[\psi_n] \rangle_{\bigotimes \mathcal{L}_N}^{n+1} \\ - \frac{1}{N} \sum_{a=2}^n \left( \langle \Theta^{(a)} \circ \widetilde{\Xi}_1^{-1}[\partial_a \psi_n] \rangle_{\bigotimes \mathcal{L}_N}^{n-1} + \langle \Theta^{(a)} \circ \widetilde{\Xi}_1^{-1}[\partial_a \psi_n] \rangle_{\mu_V \bigotimes \mathcal{L}_N}^{n-2} \right) . \quad (66) $$

**Proof** See for example [BGK16][Prop 3.2.3] done in a similar context. $\square$

## 5.2 Asymptotic expansion of linear statistics

The a priori bound on the linear statistics of Theorem 2.4, provides a starting point for obtaining the existence of their large-$N$ asymptotic expansion in powers of $N^{-1}$ up to any order through an analysis of the loop equations.

**Theorem 5.2** Let $\psi_k \in \bigcap_{m \ge 0} H^m(\mathbb{R}^k)$, then for all integer $K$, there exists a sequence $(d_a^{(k),V})_{a \ge \lceil k/2 \rceil} \in \mathbb{R}^N$ such that

$$ \langle \psi_k \rangle_k = \sum_{a=\lceil k/2 \rceil}^{K} \frac{d_a^{(k),V}(\psi_k)}{N^a} + O(N^{-(K+1)}) $$
with

$$d_1^{(1),V}[\psi_1] = P \left\langle \partial_1 \Xi^{-1}[\psi_1] \right\rangle_{\mu_V} + P \left\langle \Theta^{(2)} \circ \Xi_1^{-1} \left[ \partial_2 \mathcal{D} \circ \Xi^{-1}[\psi_1] \right] \right\rangle_{\mu_V}.$$

Furthermore, there exists a sequence $(m_{K,k}) > 0$, increasing in $K$, such that for all $k \ge 1$ and $K \ge 0$, all $\psi_k \in \bigcap_{m \ge 0} H^m(\mathbb{R}^k)$,

$$\left| \langle \psi_k \rangle_k \otimes_{\mathcal{L}_N} - \sum_{a=[k/2]}^{K} \frac{d_a^{(k),V}(\psi_k)}{N^a} \right| \le \frac{\mathcal{C}_{\text{rem}}(V, K, k)}{N^{K+1}} \mathcal{N}_{m_{K,k}}^{(k)}(\psi_k). \quad (67)$$

Above $\mathcal{N}_m^{(n)}(\psi_m) := \max (\|\psi_n\|_{W^\infty(\mathbb{R}^m)}, \|\psi_n\|_{H^n(\mathbb{R}^m)})$, while $\mathcal{C}_{\text{rem}}(V, K, k) > 0$ is a constant depending on $V$, $K$ and $k$. Finally, under the choice of potential $V_{\phi,t}$ defined in Theorem 1.3, for $\phi \in \cap_{k \ge 0} H^k(\mathbb{R})$ the map $t \in [0,1] \mapsto \mathcal{C}_{\text{rem}}(V_{\phi,t}, K, k)$ is continuous.

**Proof** Using the first loop equation given in Theorem 5.1, we get:

$$\langle \psi_1 \rangle_{\mathcal{L}_N} = \frac{P}{N} \left\langle \partial_1 \Xi^{-1}[\psi_1] \right\rangle_{\mu_V} + \frac{P}{N} \left\langle \partial_1 \Xi^{-1}[\psi_1] \right\rangle_{\mathcal{L}_N} - P \left\langle \mathcal{D} \circ \Xi^{-1}[\psi_1] \right\rangle_{\mathcal{L}_N \otimes \mathcal{L}_N}. \quad (68)$$

where we recall that $\Xi^{-1} = \Xi^{-1} \circ \mathcal{X}$ defined in 4.10. The idea is to verify the hypotheses of Theorems 2.4 for each function involved in the Dyson-Schwinger equations. By Proposition 3.5 and Theorem 3.3, 4.14, 4.15, 4.17 and 4.18 and the fact that $\psi_k \in \cap_{m \ge 0} H^m(\mathbb{R}^k)$, we're ensured that all the norms are finite and that a $n$-linear statistic will be a $O(N^{-\frac{n}{2}(1-\varepsilon)})$ where $\varepsilon > 0$ is fixed but can be chosen arbitrarily small.

We show by induction on $K$ that there exists an asymptotic expansion up to $o(N^{-K})$ for any function $\psi_k \in \cap_{m \ge 0} H^m(\mathbb{R}^k)$ for all $k \le 2K$.

For $K=1$, since the first term in (68) clearly contributes to the asymptotic expansion of $\langle \psi_1 \rangle_{\mathcal{L}_N}$ up to $o(N^{-1})$ so we focus on the two other terms. In (68), the second term is clearly a $o(N^{-1})$ since by Theorem 2.4, Theorem 4.14 and Theorem 4.15

$$\begin{aligned} \left| \langle \partial_1 \Xi^{-1}[\psi_1] \rangle_{\mathcal{L}_N} \right| &\le C_{1,\varepsilon} e^{KV} N^{-(1-\varepsilon)/2} \left( \|\partial_1 \Xi^{-1}[\psi_1]\|_{H^{1/2}(\mathbb{R})} + \|\partial_1 \Xi^{-1}[\psi_1]\|_{W_1^\infty(\mathbb{R})} \right) \\ &\le C_{1,\varepsilon} e^{KV} N^{-(1-\varepsilon)/2} \left( \mathcal{C}(\Xi_1^{-1}, H^2) + \mathcal{C}(\Xi_1^{-1}, W_2^\infty) \right) \mathcal{N}_3(\psi_1). \end{aligned}$$

To obtain the expansion of the 2-linear statistic up to $o(N^{-1})$, we will need to use the loop equation at level 2 with $\psi_2 := \mathcal{D} \circ \Xi^{-1}[\psi]$. Let $\psi_2 \in \cap_{k \ge 0} H^k(\mathbb{R}^2)$ be arbitrary for now. The level 2 equation reads:

$$\begin{aligned} \langle \psi_2 \rangle_2 \otimes_{\mathcal{L}_N} &= \frac{P}{N} \langle \partial_1 \Xi_1^{-1}[\psi_2] \rangle_{\mu_V} \otimes_{\mathcal{L}_N} + \frac{P}{N} \langle \partial_1 \Xi_1^{-1}[\psi_2] \rangle_2 \otimes_{\mathcal{L}_N} - \frac{1}{N} \langle \Theta^{(2)} \circ \Xi_1^{-1} [\partial_2 \psi_2] \rangle_{\mathcal{L}_N} \\ &\quad - P \langle \mathcal{D}_1 \circ \Xi_1^{-1}[\psi_2] \rangle_3 \otimes_{\mathcal{L}_N} - \frac{1}{N} \langle \Theta^{(2)} \circ \Xi_1^{-1} [\partial_2 \psi_2] \rangle_{\mu_V}. \end{aligned} \quad (69)$$

The first term is a $o(N^{-1})$ as a 1-linear statistic $\langle \psi \rangle_{\mathcal{L}_N}$ where $\psi(x) := \int_{\mathbb{R}} \partial_1 \Xi_1^{-1}[\psi_2](x, y)d\mu_V(y)$. This function is indeed in $H^1(\mathbb{R})$ because of Theorem 4.14

$$\|\psi\|_{H^1(\mathbb{R})}^2 \le \|\rho_V\|_{\infty} \|\partial_1 \Xi_1^{-1}[\psi_2]\|_{H^1(\mathbb{R}^2)}^2 \le \|\rho_V\|_{\infty} \mathcal{C}(\Xi_1^{-1}, H^2)^2 \|\psi_2\|_{H^3(\mathbb{R}^2)}^2$$
and in $W_1^\infty(\mathbb{R})$ by Theorem 4.15

$$ \|\psi\|_{W_1^\infty(\mathbb{R})} \le \|\partial_1 \Xi_1^{-1}[\psi_2]\|_{W_1^\infty(\mathbb{R}^2)} \le \mathcal{C}(\Xi_1^{-1}, W_2^\infty) \|\psi_2\|_{W_3^\infty(\mathbb{R}^2)}. $$

Thus by the a priori bound Theorem 2.4, we get

$$ \left| \frac{P}{N} \left\langle \partial_1 \Xi_1^{-1}[\psi_2] \right\rangle_{\mu_V} \otimes \mathcal{L}_N \right| \le \frac{PC_{1,\varepsilon} e^{K_V}}{N^{1+(1-\varepsilon)/2}} \left[ \mathcal{C}(\Xi_1^{-1}, W_2^\infty) + \|\rho_V\|_{\infty}^{1/2} \mathcal{C}(\Xi_1^{-1}, H^2) \right] \mathcal{N}_3^{(2)}(\psi_2). \quad (70) $$

The following two terms in (69) are also a $o(N^{-1})$ by the same reasons as before. By Theorem 2.4, the 3-linear statistics is a $o(N^{-1})$ for $\varepsilon > 0$ small enough. Hence, we obtain the expansion:

$$ \langle \psi \rangle_{\mathcal{L}_N} = \frac{d_1^{(1)}(\psi_1)}{N} + o(N^{-1}) \quad \text{and} \quad \langle \psi_2 \rangle_{\mathcal{L}_N \otimes \mathcal{L}_N} = \frac{d_1^{(2)}(\psi_2)}{N} + o(N^{-1}) $$

where

$$ d_1^{(1)}(\psi_1) := P \left\langle \partial_1 \Xi^{-1}[\psi_1] \right\rangle_{\mu_V} + P \left\langle \Theta^{(2)} \circ \Xi_1^{-1} \left[ \partial_2 \mathcal{D} \circ \Xi^{-1}[\psi_1] \right] \right\rangle_{\mu_V} $$

and

$$ d_1^{(2)}(\psi_2) := - \left\langle \Theta^{(2)} \circ \Xi_1^{-1} \left[ \partial_2 \psi_2 \right] \right\rangle_{\mu_V}. $$

More generally, suppose the desired expansion for $\langle \psi_k \rangle_k \otimes \mathcal{L}_N$ holds up to $o(N^{-n})$ for all $k \in [0, 2n]$ and for any function $\psi_k \in \cap_{m \ge 0} H^m(\mathbb{R}^k)$. Additionally, suppose that (67) is true for all $k \in [1, K-2]$. Then, taking a general function $\psi_{2n+2} \in \cap_{m \ge 0} H^m(\mathbb{R}^{2n+2})$, the $(2n+2)$-th equation involves the $2n+3$ linear statistic $\langle \mathcal{D}_1 \circ \Xi_1^{-1}[\psi_{2n+2}] \rangle_{2n+3} \otimes \mathcal{L}_N$ (see (66)). By Theorem 2.4, it will be a $o(N^{-(n+1)})$ for $\varepsilon$ small enough. The other terms will be either, $2n+1$-linear statistics with a $N^{-1}$ prefactor and therefore behave like $O(N^{-(n+1)})$ for $\varepsilon$ small enough, or either be $2n$-linear statistics with a prefactor $N^{-1}$. For the latter, by hypothesis, we know the asymptotic expansion up to $O(N^{-n})$, thus with the prefactor $N^{-1}$, we deduce the following expansion for $\langle \psi_{2n+2} \rangle_{2n+2} \otimes \mathcal{L}_N$

$$ \langle \psi_{2n+2} \rangle_{2n+2} \otimes \mathcal{L}_N = \frac{d_{n+1}^{(2n+2)}(\psi_{2n+2})}{N^{n+1}} + o(N^{-(n+1)}) $$

We will deduce from that, the asymptotic expansion of $\langle \psi_{2n+1} \rangle_{2n+1} \otimes \mathcal{L}_N$ for a general $\psi_{2n+1}$ belonging to $\cap_{k \ge 0} H^k(\mathbb{R}^{2n+1})$. In the $(2n+1)$-th loop equation (66), the $2n+2$ linear statistic will yield a non-trivial term of order $N^{-(n+1)}$ ie

$$ \langle \mathcal{D}_1 \circ \Xi_1^{-1}[\psi_{2n+1}] \rangle_{2n+2} \otimes \mathcal{L}_N = \frac{d_{n+1}^{(2n+2)} \left( \mathcal{D}_1 \circ \widetilde{\Xi}_1^{-1}[\psi_{2n+1}] \right)}{N^{n+1}} + o(N^{-(n+1)}). $$

Again, the $(2n+1)$-linear statistics with a prefactor $N^{-1}$ will be $o(N^{-(n+1)})$. Finally for the $2n$ and $2n-1$ linear statistics with the prefactor $N^{-1}$ appearing in the $(2n+1)$-th loop equation, we know the asymptotic expansion up to $o(N^{-(n+1)})$ by hypothesis.

To conclude on the asymptotic expansion up to $o(N^{-(n+1)})$ for the $2n$-linear statistics, just notice that for each term appearing in the $2n$ equation each term will either be a $2n+1$-linear statistics for which we know the asymptotic expansion up to $o(N^{-(n+1)})$, or a linear statistic for which we know, by hypothesis, the asymptotic expansion up to $o(N^{-n})$ (of order $2n, 2n-1$ or $2n-2$), preceded
by a factor $N^{-1}$. We can therefore conclude on the existence of the asymptotic expansion up to $o(N^{-(n+1)})$ for the 2n-linear statistics. Then applying the same arguments for $2n-1, \dots, 1$-linear statistics allows us to conclude that the induction step is established.

Finally, to conclude on (67), one just has to notice that for all $n \ge 1$ and $\psi_n \in \cap_{k \ge 0} H^k(\mathbb{R}^n)$, for all $K \ge [n/2]$,

$$ \langle \psi_n \rangle_n \otimes_{\mathcal{L}_N} - \sum_{a=[n/2]}^{K} \frac{d_a^{(n)}(\psi_n)}{N^a} = \frac{d_{K+1}^{(n)}(\psi_n)}{N^{K+1}} + \mathfrak{R}_{K+1}^{(n)}(\psi_n) \quad \text{and} \quad \mathfrak{R}_{K+1}^{(n)}(\psi_n) = o(N^{-(K+1)}). $$

Above, the remainder $\mathfrak{R}_{K+1}^{(n)}(\psi_n)$ contains all the negligible (by the a priori bound) statistics involving the operators $\Xi_1^{-1}$, $\mathcal{D}_1 \circ \Xi_1^{-1}$ and $\Theta^{(a)} \circ \Xi_1^{-1}$, thus just as in (70), by using continuity of the different operators involved in each of the statistics, there exists $m_{K,n}^{(1)} > 0$, a polynomial $Q_{K,n}^{(1)}$ in $e^{K_V}$, $\|\rho_V\|_{\infty}^{1/2}$, $(c(\Xi_1^{-1}, H^i))_{0 \le i \le m_{K,n}^{(1)}}$ $(c(\Xi_1^{-1}, W_i^{\infty}))_{0 \le i \le m_{K,n}^{(1)}}$ and $(c(\Theta^{(a)} \circ \Xi_1^{-1}, H^n))_{0 \le i \le m_{K,n}^{(1)}}$ with coefficients independent of $V$ and a constant $C(K,n) > 0$ such that

$$ |\mathfrak{R}_{K+1}^{(n)}[\psi_n]| \le \frac{C^{(1)}(K,n)}{N^{K+1}} Q_{K,n}^{(1)} \mathcal{N}_{m_{K,n}^{(1)}}^{(n)}(\psi_n). $$

To bound, $d_{K+1}^{(n)}(\psi_n)$ and extract the $V$-dependance, one just notices that it is a sum of linear statistics, involving as before the previous operators. By continuity of the operators, there exists a polynomial $Q_{K,n}^{(2)}$ in the previous operator norms and $\|\rho_V\|_{\infty}^{1/2}$ with coefficients independent of $V$ such that

$$ |d_{K+1}^{(n)}(\psi_n)| \le C^{(2)}(K,n) Q_{K,n}^{(2)} \mathcal{N}_{m_{K,n}^{(2)}}^{(n)}(\psi_n). $$

Thus setting $c_{\text{rem}}(V, K, n) = C^{(1)}(K, n)Q_{K,n}^{(1)} + C^{(2)}(K, n)Q_{K,n}^{(2)}$ and $m_{K,n} := \max(m_{K,n}^{(1)}, m_{K,n}^{(2)})$ allows us to conclude about 67. The fact that $t \in [0, 1] \mapsto c_{\text{rem}}(V_{\phi,t}, K, n)$ is continuous follows from the fact it is a polynomial in building blocks which are continuous as it is shown in Appendix B, Lemma B.10, Proposition B.8. $\square$

# 6 Parameter continuity of the equilibrium measure

We want to conclude about the asymptotic expansion of $\log \mathcal{Z}_N[V_{G,\phi}]$ for a smooth $\phi$, by inserting the asymptotic expansion of the linear statistics of Theorem 5.2 in Lemma 7.3. In order to make that step rigorous, it is necessary to prove that all the linear statistics integrated with respect to the probability measure $\mathbb{P}_N^{V_{G,\phi,t}}$ with $t \in [0, 1]$, when integrating them with respect to $t$, yield a definite and finite integral. Since all the quantities depend on $t$ through $\mu_{V_{G,\phi,t}}$, we first prove a continuity result for $t \mapsto \rho_{V_{G,\phi,t}}$. The result that we are going to prove does not depend on the specificity of the Gaussian potential so in the following, we set $V_{\phi,t} : x \mapsto V(x) + t\phi(x)$ with $V$ satisfying the assumptions 1.1.

While from the measure point of view, it is easy to show that $t \mapsto \mu_{V_{\phi,t}}$ is continuous for the weak topology of measures, it is not sufficient to deduce the continuity of the quantities involved in our problem. Indeed, in the controls we showed in Section 4 quantities like $L^{\infty}$-norm of derivatives of $\rho_{V_t}$ and $C_L$, it will be necessary to show that $\|\rho_{V_{\phi,t}} - \rho_{V_{\phi,t_0}}\|_{W_i^{\infty}(\mathbb{R})} \xrightarrow{t \to t_0} 0$ for all $t_0 \in [0, 1]$.
## 6.1 Setting for Banach fixed-point theorem

Let $\phi \in C^{\infty}(\mathbb{R})$, such that for all $k \in \mathbb{N}$, $\phi^{(k)} \in L^2(\mathbb{R})$, let $t_0 \in [0, 1]$, we define the function $u_t$ by

$$ \rho_{V_{\phi,t}} = (1 + \delta t u_t) \rho_{V_{\phi,t_0}} \quad \text{ie} \quad u_t = \frac{\rho_{V_{\phi,t}} - \rho_{V_{\phi,t_0}}}{\delta t} \frac{1}{\rho_{V_{\phi,t_0}}} \quad (71) $$

where $t \neq t_0$ and $\delta t := t - t_0$. We will show, by Banach fixed-point theorem, that $x \mapsto u_t(x) \in C^{\infty}(\mathbb{R})$, by Lemma 2.1, is the unique fixed-point of a $t$ continuous operator. This will allow us to deduce that $t \mapsto u_t$ is continuous for the $\bigcap_{k \in \mathbb{N}} W_k^{\infty}$-norm. The continuity of $t \mapsto \rho_{V_t}$ will then follow.

In order to construct the operator of interest, we start with the following lemma

**Lemma 6.1** Let $t, t_0 \in [0, 1]$,

$$ \lambda_{V_{\phi,t}} = \lambda_{V_{\phi,t_0}} + \delta t \int_{\mathbb{R}} \phi(x) d\mu_{V_{\phi,t_0}}(x) - 2P \delta t \iint_{\mathbb{R}^2} \log|x-y|u_t(x)d\mu_{V_{\phi,t_0}}(x)d\mu_{V_{\phi,t_0}}(y) \\ + \int_{\mathbb{R}} [\log(1 + \delta t u_t(x)) - \delta t u_t(x)] d\mu_{V_{\phi,t_0}}(x). \quad (72) $$

Here $\lambda_{V_{\phi,t}}$ denotes the constant appearing in (4) with potential $V_{\phi,t}$.

**Proof** We integrate with respect to $\mu_{V_{\phi,t_0}}$ (4) to get

$$ \lambda_{V_{\phi,t}} = \int_{\mathbb{R}} V_{\phi,t_0}(x) d\mu_{V_{\phi,t_0}}(x) + \delta t \int_{\mathbb{R}} \phi(y) d\mu_{V_{\phi,t_0}}(y) - 2P \iint_{\mathbb{R}^2} \log|x-y| d\mu_{V_{\phi,t}}(x) d\mu_{V_{\phi,t_0}}(y) \\ + \int_{\mathbb{R}} \log \rho_{V_{\phi,t}}(x) d\mu_{V_{\phi,t_0}}(x). $$

After using the fact that $\int_{\mathbb{R}} u_t d\mu_{V_{\phi,t_0}} = 0$, that $\rho_{V_{\phi,t}} = (1 + \delta t u_t)\rho_{V_{\phi,t_0}}$ and the characterization (4) of $\mu_{V_{\phi,t_0}}$, this yields the result. $\square$

To show that $u_t$ is a fixed point of a $t$-continuous operator, we need to invert and control the operator $\mathcal{T} := \mathcal{L} \circ \mathcal{A}^{-1}$ (these operators were introduced in Definition 4.1) which will appear naturally when comparing $\rho_{V_{\phi,t}}$ to $\rho_{V_{\phi,t_0}}$.

**Proposition 6.2** We define the operator $\mathcal{T}$ by $\mathcal{T}[v] := v - \mathcal{K}[v]$ for all $v \in L^2(\mu_{V_{\phi,t_0}})$, where

$$ \mathcal{K}[v](x) := 2P \int_{\mathbb{R}} k(x, y)v(y)\rho_{V_{\phi,t_0}}(y)dy $$

and

$$ k(x, y) := \left( \log \frac{|x - y|}{1 + |x|} - \int_{\mathbb{R}} \log \frac{|z - y|}{1 + |z|} \rho_{V_{\phi,t_0}}(z) dz \right). $$

$\mathcal{T}: L^2(\mu_{V_{\phi,t_0}}) \to L^2(\mu_{V_{\phi,t_0}})$ is bijective and for all $n \geq 0$, $\mathcal{T}[W_n^{\infty}(\mathbb{R})] = W_n^{\infty}(\mathbb{R})$. Finally, for all $n \in \mathbb{N}$, there exists $C_{\mathcal{T},n} > 0$ such that for any $v \in W_n^{\infty}(\mathbb{R})$,

$$ \|\mathcal{T}^{-1}[v]\|_{W_n^{\infty}(\mathbb{R})} \leq C_{\mathcal{T},n} \|v\|_{W_n^{\infty}(\mathbb{R})}. \quad (73) $$
**Proof** It was shown in [DGM23] that $\mathcal{T}[v] = -\mathcal{L} \circ \mathcal{A}^{-1}[v]$. $\mathcal{A}$ and $\mathcal{L}$ are unbounded operators on $\mathcal{H}$, it was also shown there that

$$ \mathcal{A} : \mathcal{D}(\mathcal{A}) := \{ \phi \in \mathcal{H}, \mathcal{A}[\phi] \in \mathcal{H} \} \to \mathcal{H} \quad \text{and} \quad \mathcal{L} : \mathcal{D}(\mathcal{A}) \to \mathcal{H} $$

are bijective, thus so is $\mathcal{T} : \mathcal{H} \to \mathcal{H}$.

The fact that $k$ verifies $\|k(x, y)\|_{L^2(\mu_{V_{\phi,t_0}} \otimes \mu_{V_{\phi,t_0}})} < +\infty$ implies that $\mathcal{K} : L^2(\mu_{V_{\phi,t_0}}) \to L^2(\mu_{V_{\phi,t_0}})$ is an Hilbert-Schmidt operator thus compact and so $\mathcal{T}$ is a Fredholm operator. We now show that the kernel of $\mathcal{T}$ is trivial. Let $v \in L^2(\mu_{V_{\phi,t_0}})$ such that $\mathcal{T}[v] = 0$ so $v = \mathcal{K}[v]$. The RHS is in $H^1(\mu_{V_{\phi,t_0}})$, indeed we have

$$ \mathcal{K}[v]'(x) = -\mathcal{H}[v\rho_{V_{\phi,t_0}}](x) - \frac{\operatorname{sgn}(x)}{1+|x|} \int_{\mathbb{R}} v(y)d\mu_{V_{\phi,t_0}}(y) \in L^2(\mu_{V_{\phi,t_0}}). $$

Moreover since $\int_{\mathbb{R}} \mathcal{K}[v]d\mu_{V_{\phi,t_0}} = 0$, we conclude that $v \in \mathcal{H}$. We can now conclude that $v = 0$ by the bijectivity of $\mathcal{T}$ on $\mathcal{H}$. Finally, by Fredhom alternative, $\mathcal{T}$ is invertible on $L^2(\mu_{V_{\phi,t_0}})$ since it is injective.

We now prove that for all $n \in \mathbb{N}$, $\mathcal{T}[W_n^\infty(\mathbb{R})] = W_n^\infty(\mathbb{R})$. We proceed by induction. For $n = 0$, let $f \in L^\infty(\mathbb{R}) \subset L^2(\mu_{V_{\phi,t_0}})$. There exists a unique $v \in L^2(\mu_{V_{\phi,t_0}})$ such that $\mathcal{T}[v] = f$ so $v = f + \mathcal{K}[v]$ but since $f$ and $\mathcal{K}[v]$ are bounded, so is $v \in L^\infty(\mathbb{R})$. Reciprocally, if $v \in L^\infty(\mathbb{R})$ so is $\mathcal{T}[v]$, hence $\mathcal{T}[L^\infty(\mathbb{R})] = L^\infty(\mathbb{R})$. Finally let $v \in L^\infty(\mathbb{R})$,

$$ \|\mathcal{T}[v]\|_{L^\infty(\mathbb{R})} \le \left( 1 + 2P \max_{x \in \mathbb{R}} \int_{\mathbb{R}} |k(x, y)| d\mu_{V_{\phi,t_0}}(y) \right) \|v\|_{L^\infty(\mathbb{R})}. \quad (74) $$

Now suppose $\mathcal{T}[W_n^\infty(\mathbb{R})] = W_n^\infty(\mathbb{R})$ is true and let's show it for $n+1$. Let $f \in W_{n+1}^\infty(\mathbb{R}) \subset W_n^\infty(\mathbb{R})$, so by hypothesis, there exists $v \in W_n^\infty(\mathbb{R})$ such that

$$ \begin{aligned} f^{(n)}(x) = v^{(n)}(x) + 2P (\log(1 + |\cdot|))^{(n)}(x) & \int_{\mathbb{R}} v(y) d\mu_{V_{\phi,t_0}}(y) - 2P \int_{\mathbb{R}} \log|x - y|(v\rho_{V_{\phi,t_0}})^{(n)}(y) dy \\ & + 2P \delta_{n,0} \iint_{\mathbb{R}^2} \frac{\log|z - y|}{1 + |z|} v(y) d\mu_{V_{\phi,t_0}}(y) d\mu_{V_{\phi,t_0}}(z). \end{aligned} \quad (75) $$

We deduce that $v^{(n)}$ is differentiable of derivative

$$ v^{(n+1)}(x) = f^{(n+1)}(x) - 2P (\log(1 + |\cdot|))^{(n+1)}(x) \int_{\mathbb{R}} v(y) d\mu_{V_{\phi,t_0}}(y) - 2P \mathcal{H} \left[ (v\rho_{V_{\phi,t_0}})^{(n)} \right](x), \quad (76) $$

where $(\log(1 + |\cdot|))^{(n)}(x) = \frac{n!\operatorname{sgn}(x)}{(1 + |x|)^n}$ if $n$ is odd and $\frac{n!}{(1 + |x|)^n}$ if $n$ is even. Since the two first terms in the RHS of (76) are clearly bounded, we just have to show that $\mathcal{H}[(v\rho_{V_{\phi,t_0}})^{(n)}] \in H^1(\mathbb{R})$. By boundedness of $f^{(n+1)}$ and $v^{(i)}$ for all $i \le n$, we have

$$ \begin{aligned} (v\rho_{V_{\phi,t_0}})^{(n+1)} &= \rho_{V_{\phi,t_0}} \left( v^{(n+1)} + \sum_{k=0}^{n} \binom{n+1}{k} \frac{\rho_{V_{\phi,t_0}}^{(n+1-k)}}{\rho_{V_{\phi,t_0}}} v^{(k)} \right) \\ &= \rho_{V_{\phi,t_0}} \left( f^{(n+1)} - 2P (\log(1 + |\cdot|))^{(n+1)} \int_{\mathbb{R}} v(y) d\mu_{V_{\phi,t_0}}(y) - 2P \mathcal{H} \left[ (v\rho_{V_{\phi,t_0}})^{(n)} \right] \right. \\ & \qquad \left. + \sum_{k=0}^{n} \binom{n+1}{k} \frac{\rho_{V_{\phi,t_0}}^{(n+1-k)}}{\rho_{V_{\phi,t_0}}} v^{(k)} \right) \in L^2(\mathbb{R}). \end{aligned} $$
Thus, it holds that $\mathcal{H}[(v\rho_{V_{\phi,t_0}})^{(n)}] \in H^1(\mathbb{R})$ and that it is bounded. Hence it proves that $v^{(n+1)} \in L^\infty(\mathbb{R})$ and hence $W_{n+1}^\infty(\mathbb{R}) \subset \mathcal{T}[W_{n+1}^\infty(\mathbb{R})]$. Conversely, if $v \in W_n^\infty(\mathbb{R})$, then $f \in W_n^\infty(\mathbb{R})$ by hypothesis and just as before, we show that (76) holds. We conclude that $f^{(n+1)} \in L^\infty(\mathbb{R})$ again by showing that $\mathcal{H}[(v\rho_{V_{\phi,t_0}})^{(n)}]$ is bounded by the fact that $v \in W_{n+1}^\infty(\mathbb{R})$. This establishes that $W_{n+1}^\infty(\mathbb{R}) = \mathcal{T}[W_{n+1}^\infty(\mathbb{R})]$.

Thus for all $n \in \mathbb{N}$, $\mathcal{T}: W_n^\infty(\mathbb{R}) \to W_n^\infty(\mathbb{R})$ is a bijective operator. Furthermore, it is a bounded operator by the fact that for all $1 \le i \le n$, there exists $C > 0$ such that for all $v \in W_n^\infty(\mathbb{R})$, by Leibniz formula

$$ \begin{aligned} |\mathcal{T}[v]^{(i)}(x)| \le \|v^{(i)}\|_{L^\infty(\mathbb{R})} + 2P \|(\log(1+|\cdot|))^{(i)}\|_{L^\infty(\mathbb{R})} \|v\|_{L^\infty(\mathbb{R})} \\ + 2P \|v\|_{W_i^\infty(\mathbb{R})} \sum_{k=0}^{i} \binom{i}{k} \sup_{z \in \mathbb{R}} \int_{\mathbb{R}} \left| \log \frac{|z-y|}{1+|z|} \right| \left| \frac{\rho_{V_{\phi,t_0}}^{(k)}}{\rho_{V_{\phi,t_0}}}(y) \right| d\mu_{V_{\phi,t_0}}(y) \le C \|v\|_{W_i^\infty(\mathbb{R})}. \end{aligned} $$

Above we used (75) and the fact that $\int_{\mathbb{R}} (v\rho_{V_{\phi,t_0}})^{(i)}(y)dy = 0$ so

$$ \begin{aligned} \int_{\mathbb{R}} \log|x-y|(v\rho_{V_{\phi,t_0}})^{(i)}(y)dy &= \int_{\mathbb{R}} \log|x-y|(v\rho_{V_{\phi,t_0}})^{(i)}(y)dy - \log(1+|x|) \int_{\mathbb{R}} (v\rho_{V_{\phi,t_0}})^{(i)}(y)dy \\ &= \int_{\mathbb{R}} \log \frac{|x-y|}{1+|x|} (v\rho_{V_{\phi,t_0}})^{(i)}(y)dy. \end{aligned} $$

Thus we conclude that $\sup_{i \in [1,n]} \|\mathcal{T}[v]^{(i)}\|_{L^\infty(\mathbb{R})} \le C \|v\|_{W_n^\infty(\mathbb{R})}$. The bound on $\|\mathcal{T}[v]\|_{L^\infty(\mathbb{R})}$ was shown in (74). We finally conclude that $\mathcal{T}: W_n^\infty(\mathbb{R}) \to W_n^\infty(\mathbb{R})$ is bounded bijective between Banach spaces and by Banach isomorphism theorem so is $\mathcal{T}^{-1}$, this establishes (73) and completes the proof. $\square$

**Remark 6.3** An explicit expression for $\mathcal{T}^{-1}$ is available Fredholm determinant theory for invertible Hilbert-Schmidt operators, see [GGK12][Section XII]. For all $v \in L_0^2(\mu_{V_{\phi,t_0}})$,

$$ \begin{aligned} \mathcal{T}^{-1}[v](x) := v(x) \\ + \frac{1}{\det(I-K)} \sum_{n \ge 1} \frac{(-1)^n}{n!} \int_{\mathbb{R}^{n+1}} \left| \begin{array}{cccc} k(x,s) & k(x,t_1) & \dots & k(x,t_n) \\ k(t_1,s) & 0 & \dots & k(t_1,t_n) \\ \vdots & \vdots & \ddots & \vdots \\ k(t_n,s) & k(t_n,t_1) & \dots & 0 \end{array} \right| v(s) d\mu_{V_{\phi,t_0}}(s) \prod_{i=1}^{n} d\mu_{V_{\phi,t_0}}(t_i). \end{aligned} \quad (77) $$

Above $\det$ stands for the 2-determinant. This formula was established in [DGM23][Theorem 6.11]

We are now able to show that $u_t$ is a fixed point of a certain operator. We recall that $\phi$ was introduced in the beginning of subsection 6.1.

**Proposition 6.4** For all $t \in [0,1]$, $u_t$ defined in (71) is the unique measurable function such that $\int_{\mathbb{R}} u_t d\mu_{V_{\phi,t_0}} = 0$ and which satisfies:

$$ u_t = \mathcal{T}^{-1} \circ \mathcal{V}_t[u_t] $$
where $\mathcal{V}_t[u] := -\phi + \int_{\mathbb{R}} \phi(y)d\mu_{V_{\phi,t_0}}(y) + \delta t \mathcal{U}_t[u]$, and

$$ \mathcal{U}_t[v](x) := \left( -\phi(x) + \int_{\mathbb{R}} \phi(y)d\mu_{V_{\phi,t_0}}(y) + \mathcal{K}[v](x) + \int_{\mathbb{R}} \frac{\log(1+\delta tv(y)) - \delta tv(y)}{\delta t} d\mu_{V_{\phi,t_0}}(y) \right)^2 \\ \times \int_0^1 (1-s)ds \exp \left\{ s\delta t \left( -\phi(x) + \int_{\mathbb{R}} \phi(y)d\mu_{V_{\phi,t_0}}(y) + \mathcal{K}[v](x) \right) \right. \\ \left. + \int_{\mathbb{R}} \frac{\log(1+\delta tv(y)) - \delta tv(y)}{\delta t} d\mu_{V_{\phi,t_0}}(y) \right\} + \int_{\mathbb{R}} \frac{\log(1+\delta tv(y)) - \delta tv(y)}{(\delta t)^2} d\mu_{V_{\phi,t_0}}(y). $$

**Proof** Lemma 6.1 allows one to substitute $\lambda_{V_{\phi,t}}$ in the representation for $\rho_{V_{\phi,t}}$ by (72) hence leading to

$$ \rho_{V_{\phi,t}} = (1+\delta t u_t)\rho_{V_{\phi,t_0}} = \exp \left( -V_{\phi,t_0} - 2P U^{\rho_{V_{\phi,t_0}}} + \lambda_{V_{\phi,t_0}} - \delta t \phi + \delta t \int_{\mathbb{R}} \phi(y) d\mu_{V_{\phi,t_0}}(y) - 2P \delta t U^{u_t \rho_{V_{\phi,t_0}}} \right. \\ \left. - 2P \delta t \iint_{\mathbb{R}^2} \log|y-z| u_t(z) d\mu_{V_{\phi,t_0}}(z) d\mu_{V_{\phi,t_0}}(y) + \int_{\mathbb{R}} \left[ \log(1+\delta t u_t(y)) - \delta t u_t(y) \right] d\mu_{V_{\phi,t_0}}(y) \right). $$

Recognizing $\rho_{V_{\phi,t_0}}$ via the first three terms in the exponential, $u_t$ has to satisfy the following relation for all $x \in \mathbb{R}$,

$$ 1 + \delta t u_t(x) = \exp \left\{ \delta t \left( -\phi(x) + \int_{\mathbb{R}} \phi(y) d\mu_{V_{\phi,t_0}}(y) + 2P \int_{\mathbb{R}} \log \frac{|x-y|}{1+|x|} u_t(y) d\mu_{V_{\phi,t_0}}(y) \right) \right. \\ \left. - 2P \iint_{\mathbb{R}^2} \log \frac{|y-z|}{1+|y|} u_t(z) d\mu_{V_{\phi,t_0}}(z) d\mu_{V_{\phi,t_0}}(y) + \frac{1}{\delta t} \int_{\mathbb{R}} \left[ \log(1+\delta t u_t) - \delta t u_t \right] d\mu_{V_{\phi,t_0}}(y) \right\}. $$

Above, we have used that $-U^{u_t \rho_{V_{\phi,t_0}}}(x) = \int_{\mathbb{R}} \log \frac{|x-y|}{1+|x|} u_t(y) d\mu_{V_{\phi,t_0}}(y)$ which is justified by the fact that $\int_{\mathbb{R}} u(y) d\mu_{V_{\phi,t_0}}(y) = 0$. Conversely, any $u$ such that $\int_{\mathbb{R}} u(y) d\mu_{V_{\phi,t_0}}(y) = 0$ and satisfying the previous relation, verifies for all $x \in \mathbb{R}$,

$$ V_t(x) + 2P U^w(x) + \log w(x) = \int_{\mathbb{R}} \left\{ V_t(y) + 2P U^w(y) + \log w(y) \right\} d\mu_{V_{\phi,t_0}}(y) $$

where we have set $w := (1 + \delta t u)\rho_{V_{\phi,t_0}}$. Because of this equation, $w$ can be written in exponential form as in (5), it is thus positive and of mass 1 which makes $d\mu(x) := w(x)dx$ a probability measure which satisfies the equation characterizing $\mu_{V_{\phi,t}}$, hence, by unicity of the solution of (4), $\mu_{V_{\phi,t}} = \mu$ and thus $u = u_t$.

We now expand exp into its Taylor-integral series of order 2, ie $e^x = 1 + x + x^2 \int_0^1 (1-s)e^{sx}ds$. By using that

$$ \mathcal{K}[v] = 2P \int_{\mathbb{R}} \log \frac{|x-y|}{1+|x|} u(y) d\mu_{V_{\phi,t_0}}(y) - 2P \iint_{\mathbb{R}^2} \log \frac{|y-z|}{1+|y|} v(z) d\mu_{V_{\phi,t_0}}(z) d\mu_{V_{\phi,t_0}}(y) $$
and $\mathcal{T}[v] = v - \mathcal{K}[v]$, we get:

$$ \mathcal{T}[u_t](x) = -\phi(x) + \int_{\mathbb{R}} \phi d\mu_{V_{\phi,t_0}} \\ + \delta t \left[ \left( -\phi(x) + \int_{\mathbb{R}} \phi d\mu_{V_{\phi,t_0}} + \mathcal{K}[u_t](x) + \frac{1}{\delta t} \int_{\mathbb{R}} [\log(1 + \delta t u_t) - \delta t u_t] d\mu_{V_{\phi,t_0}} \right)^2 \right. \\ \left. \times \int_0^1 \exp \left( s\delta t \left( -\phi(x) + \int_{\mathbb{R}} \phi d\mu_{V_{\phi,t_0}} + \mathcal{K}[u_t](x) + \int_{\mathbb{R}} \frac{\log(1 + \delta t u_t) - \delta t u_t}{\delta t} d\mu_{V_{\phi,t_0}} \right) \right) (1-s)ds \right. \\ \left. + \int_{\mathbb{R}} \frac{\log(1 + \delta t u_t) - \delta t u_t}{(\delta t)^2} d\mu_{V_{\phi,t_0}} \right]. $$

We next use the invertibility of $\mathcal{T}$ to conclude. $\square$

The next theorem shows that for each $t$ sufficiently close to $t_0$, $\mathcal{T}^{-1} \circ \mathcal{V}_t$ is contractive on a ball of fixed radius. Let $n \ge 0$, denote for all $R > 0$, $\mathcal{B}_n(0, R)$ the closed ball of radius $R_n$ for the $W_n^\infty(\mathbb{R})$-norm.

**Theorem 6.5** For all $\varepsilon_n > 0$ small enough, there exists $R_n > 0$, such that for all $t \in ]t_0 - \varepsilon_n, t_0 + \varepsilon_n[$, the operator

$$ \mathcal{T}^{-1} \circ \mathcal{V}_t : \left( \overline{\mathcal{B}_n(0, R_n)}, \|\cdot\|_{W_n^\infty(\mathbb{R})} \right) \to \left( \overline{\mathcal{B}_n(0, R_n)}, \|\cdot\|_{W_n^\infty(\mathbb{R})} \right) $$

is well-defined and continuous. Furthermore it is contractive, ie there exists $k_n \in ]0, 1[$, such that for all $v, w \in \overline{\mathcal{B}_n(0, R_n)}$,

$$ \| \mathcal{T}^{-1} \circ \mathcal{V}_t[v] - \mathcal{T}^{-1} \circ \mathcal{V}_t[w] \|_{W_n^\infty(\mathbb{R})} \le k_n \|v - w\|_{W_n^\infty(\mathbb{R})}. $$

Moreover $k_n$ is independent of $t$ on $]t_0 - \varepsilon_n, t_0 + \varepsilon_n[$ for $\varepsilon_n > 0$ small enough.

**Proof** Let $\varepsilon > 0$, $t \in ]t_0 - \varepsilon, t_0 + \varepsilon[$, $v \in \overline{\mathcal{B}_0(0, R)}$ for an arbitrary $0 < R < \frac{1}{\delta t}$ (because otherwise the term $\int_{\mathbb{R}} \log(1 + \delta t v) \rho_{V_{\phi,t_0}}$ in $\mathcal{V}_t[v]$ might be ill-defined). We first show that $\|\mathcal{V}_t[v]\|_{L^\infty(\mathbb{R})} < +\infty$. First, by Taylor-Lagrange inequality, we deduce that

$$ \left| \int_{\mathbb{R}} \frac{\log[1 + \delta t v(y)] - \delta t v(y)}{\delta t} d\mu_{V_{\phi,t_0}}(y) \right| \le \frac{\|v\|_{L^\infty(\mathbb{R})}^2 \delta t}{2} \sup_{x \in [-\|v\|_\infty, \|v\|_\infty]} \frac{1}{(1 + \delta t x)^2} \le \frac{R^2 \delta t}{2(1 - \delta t R)^2}. $$

By recalling the definition of $\mathcal{V}_t$ in Proposition 6.4 and using the convexity of $x \mapsto x^2$, we get:

$$ \begin{aligned} \|\mathcal{V}_t[v]\|_{L^\infty(\mathbb{R})} \le \delta t \left[ 3 \left( 4 \|\phi\|_{L^\infty(\mathbb{R})}^2 + R^2 \left\| \int_{\mathbb{R}} \left| \log \frac{|\cdot - y|}{1 + |\cdot|} \right| d\mu_{V_{\phi,t_0}}(y) dy \right\|_{L^\infty(\mathbb{R})}^2 + \delta t^2 \frac{R^4}{4(1 - \delta t R)^4} \right) \right. \\ \left. \times \exp \left\{ \delta t \left( 2 \|\phi\|_{L^\infty(\mathbb{R})} + R \left\| \int_{\mathbb{R}} \left| \log \frac{|\cdot - y|}{1 + |\cdot|} \right| d\mu_{V_{\phi,t_0}}(y) dy \right\|_{L^\infty(\mathbb{R})} + \delta t \frac{R^2}{2(1 - \delta t R)^2} \right) \right\} \right] \\ + \frac{R^2}{2(1 - \delta t R)^2} \right] \\ + 2 \|\phi\|_{L^\infty(\mathbb{R})}. \end{aligned} $$
The RHS is of the form $2\|\phi\|_{\infty} + \delta t g_0(\delta t)$ where $g_0$ is a positive function and with these notations

$$ \|\mathcal{T}^{-1} \circ \mathcal{V}_t[v]\|_{L^{\infty}(\mathbb{R})} \leq C_{\mathcal{T},0} (2\|\phi\|_{L^{\infty}(\mathbb{R})} + \delta t g_0(\delta t)). $$

Therefore, by choosing $t$ such that $|t - t_0| < \varepsilon_0$ for $\varepsilon_0 > 0$ small enough, there exists $R_0 > 0$ such that $2\|\phi\|_{\infty} C_{\mathcal{T},0} < R_0 < \frac{1}{\varepsilon_0}$ and

$$ \mathcal{T}^{-1} \circ \mathcal{V}_t(\overline{\mathcal{B}_0(0, R_0)}) \subset \overline{\mathcal{B}_0(0, R_0)}. $$

This makes the operator $\mathcal{T}^{-1} \circ \mathcal{V}_t : (\overline{\mathcal{B}_n(0, R_n)}, \|\cdot\|_{W_n^{\infty}(\mathbb{R})}) \to (\overline{\mathcal{B}_n(0, R_n)}, \|\cdot\|_{W_n^{\infty}(\mathbb{R})})$ well-defined for all $|t - t_0| < \varepsilon_0$. For the contractivity, let $u, v \in \overline{\mathcal{B}(0, R_0)}$, we get by Proposition 6.2,

$$ \|\mathcal{T}^{-1}[\mathcal{V}_t[u] - \mathcal{V}_t[v]]\|_{L^{\infty}(\mathbb{R})} \leq C_{\mathcal{T},0} \|\mathcal{U}_t[u] - \mathcal{U}_t[v]\|_{L^{\infty}(\mathbb{R})}. $$

where $\mathcal{U}_t$ was defined in Proposition 6.4. We now want to control $|\mathcal{U}_t[u](x) - \mathcal{U}_t[v](x)|$. We have, by decomposing the sum

$$ |\mathcal{U}_t[u](x) - \mathcal{U}_t[v](x)| \leq \delta t (\Delta_1 + \Delta_2(x)\mathbb{E}[u](x) + \Delta_3(x)\mathfrak{h}[v](x)^2) $$

where

$$ \mathfrak{h}[w](x) := -\phi(x) + \int_{\mathbb{R}} \phi d\mu_{V_{\phi,t_0}} + \mathcal{K}[w](x) + \int_{\mathbb{R}} \frac{\log(1 + \delta t w) - \delta t w}{\delta t} d\mu_{V_{\phi,t_0}} $$

$$ \mathbb{E}[w](x) := \int_{0}^{1} \exp \left[ s \delta t \mathfrak{h}[w](x) \right] (1-s) ds $$

$$ \Delta_1 := \int_{\mathbb{R}} \left| \frac{\log(1 + \delta t u) - \delta t u - \log(1 + \delta t v) + \delta t v}{(\delta t)^2} \right| d\mu_{V_{\phi,t_0}} $$

$$ \Delta_2(x) := |\mathfrak{h}[u](x)^2 - \mathfrak{h}[v](x)^2| $$

$$ \Delta_3(x) := |\mathbb{E}[u](x) - \mathbb{E}[v](x)|. $$

First, Taylor Lagrange inequality leads to:

$$ \Delta_1 \leq \sup_{y \in [-R_0, R_0]} \frac{|y|}{1 + \delta t y} \|u - v\|_{L^{\infty}(\mathbb{R})} \leq \frac{R_0}{1 - \delta t R_0} \|u - v\|_{L^{\infty}(\mathbb{R})}. $$

Furthermore, by using $a^2 - b^2 = (a + b)(a - b)$, that $\mathfrak{h}[u]$ and $\mathfrak{h}[v]$ are bounded

$$ \begin{aligned} \Delta_2 &\leq 2 \max (\|\mathfrak{h}[u]\|_{L^{\infty}(\mathbb{R})}, \|\mathfrak{h}[v]\|_{L^{\infty}(\mathbb{R})}) \left[ |\mathcal{K}[u - v](x)| + \delta t \Delta_1 \right] \\ &\leq \left( 4 \|\phi\|_{L^{\infty}(\mathbb{R})} + 8 P R_0 \left\| \int_{\mathbb{R}} \left| \log \frac{|\cdot - y|}{1 + |\cdot|} \right| d\mu_{V_{\phi,t_0}}(y) \right\|_{L^{\infty}(\mathbb{R})} + \frac{\delta t R_0^2}{(1 - \delta t R_0)^2} \right) \|u - v\|_{L^{\infty}(\mathbb{R})} \\ &\quad \times \left( \frac{R_0 \delta t}{1 - \delta t R_0} + 4 P \left\| \int_{\mathbb{R}} \left| \log \frac{|\cdot - y|}{1 + |\cdot|} \right| \rho_{V_{\phi,t_0}}(y) dy \right\|_{L^{\infty}(\mathbb{R})} \right) \leq C(R_0) \|u - v\|_{L^{\infty}(\mathbb{R})} \end{aligned} $$

Similarly, there exists $C(R_0) > 0$ such that:

$$ \Delta_3 \|\mathfrak{h}[v]\|_{L^{\infty}(\mathbb{R})} \leq C(R_0) \delta t \|u - v\|_{L^{\infty}(\mathbb{R})} $$
which finally leads to the existence of $C_0 > 0$ such that:

$$ \|\mathcal{U}_t[u] - \mathcal{U}_t[v]\|_{L^\infty(\mathbb{R})} \le C_0 \delta t \|u - v\|_{L^\infty(\mathbb{R})}. $$

We now choose $\varepsilon_0$ again small enough such that $k_0 := \delta t C_0 C_{\mathcal{T},0} < 1$, this concludes the proof that $\mathcal{T}^{-1} \circ \mathcal{V}_t$ is contractive on $\overline{\mathcal{B}_0(0, R_0)}$ with contractivity constant $k_0$.

To get the contractivity property for $\mathcal{T}^{-1} \circ \mathcal{V}_t$ on $W_n^\infty(\mathbb{R})$, we adapt a similar strategy. Let $u \in \overline{\mathcal{B}_n(0, R)}$ with $\delta t R < 1$, by Proposition 6.2,

$$ \|\mathcal{T}^{-1}[\mathcal{V}_t[u]]\|_{W_n^\infty(\mathbb{R})} \le C_{\mathcal{T},n} \|\mathcal{V}_t[u]\|_{W_n^\infty(\mathbb{R})}. $$

Furthermore, it is clear that every term appearing in the definition of $\mathcal{V}_t[u]$ belongs to $W_n(\mathbb{R})$, thus by the same argument as before there exists a positive function $g_n$ such that,

$$ \|\mathcal{T}^{-1} \circ \mathcal{V}_t[u]\|_{W_n(\mathbb{R})} \le C_{\mathcal{T},n} (2\|\phi\|_{W_n^\infty(\mathbb{R})} + \delta t g_n(R)). $$

We conclude just as before that by taking $\delta t$ small enough, $\mathcal{T}^{-1} \circ \mathcal{V}_t : \overline{\mathcal{B}_n(0, R)} \to \overline{\mathcal{B}_n(0, R)}$ is well-defined. Finally, just as before since for all $u, v \in W_n^\infty(\mathbb{R})$

$$ \begin{aligned} \mathcal{U}_t[u]^{(n)}(x) &= \delta t \sum_{k=0}^{n} \binom{n}{k} (\mathfrak{h}[u]^2)^{(k)} (x) \mathbf{E}[u]^{(n-k)}(x) \\ &= \delta t \sum_{0 \le i \le k \le n} \binom{n}{k} \binom{k}{i} \mathfrak{h}[u]^{(i)}(x) \mathfrak{h}[u]^{(k-i)}(x) \mathbf{E}[u]^{(n-k)}(x). \end{aligned} $$

Moreover, by the same controls as before it is easy to derive that for all $0 \le i \le k \le n$, for all $u, v \in \overline{\mathcal{B}_n(0, R_n)}$,

$$ \|\mathfrak{h}[u]^{(i)}\mathfrak{h}[u]^{(k-i)}\mathbf{E}[u]^{(n-k)} - \mathfrak{h}[v]^{(i)}\mathfrak{h}[v]^{(k-i)}\mathbf{E}[v]^{(n-k)}\|_{L^\infty(\mathbb{R})} \le C(R_n, n, i, k)\|u - v\|_{W_n^\infty(\mathbb{R})}. $$

This is enough to conclude that

$$ \|\mathcal{T}^{-1} \circ \mathcal{V}_t[u] - \mathcal{T}^{-1} \circ \mathcal{V}_t[v]\|_{W_n^\infty(\mathbb{R})} \le \delta t C_{\mathcal{T},n} C(R_n, n, i, k) \|u - v\|_{W_n^\infty(\mathbb{R})}. $$

Finally, by taking $\delta t$ small enough, we conclude that $\mathcal{T}^{-1} \circ \mathcal{V}_t$ is contractive on $\overline{\mathcal{B}_n(0, R_n)}$ with contractivity constant $k_n := \delta t C_{\mathcal{T},n} C(R_n, n, i, k) < 1$. $\square$

**Remark 6.6** Note that the definition of $u_t$ as an element of $W_n^\infty(\mathbb{R})$ depends on $t_0$ and that we only proved the characterization of $u_t$ as a fixed point for $t \in [t_0 - \varepsilon_n, t_0 + \varepsilon_n[\setminus\{t_0\}]$ with $\varepsilon_n > 0$ small enough (we stress that we successively lowered $\delta t$ when increasing $i$.) Furthermore, since for all $v \in L^\infty(\mathbb{R})$

$$ \mathcal{T}^{-1} \circ \mathcal{V}_{t_0}[v] = -\phi + \int_{\mathbb{R}} \phi d\mu_{\mathcal{V}_{\phi, t_0}}, $$

then we can set $u_{t_0} := -\phi + \int_{\mathbb{R}} \phi d\mu_{\mathcal{V}_{\phi, t_0}}$ which is obviously the unique fixed point in $\bigcap_{n \in \mathbb{N}} W_n^\infty(\mathbb{R})$ of $\mathcal{T}^{-1} \circ \mathcal{V}_{t_0}$.
## 6.2 Regularity of the equilibrium measure

We now prove the continuity of $t \mapsto u_t \in W_n^\infty(\mathbb{R})$.

**Lemma 6.7** Let $n \in \mathbb{N}$, $\varepsilon_n > 0$ and $R_n > 0$ be as in Theorem 6.5. For all $v \in \overline{\mathcal{B}_n(0, R_n)}$, for all $(t_p)_p \in (]t_0 - \varepsilon_n, t_0 + \varepsilon_n[)^N$ such that $t_p \underset{p \to \infty}{\longrightarrow} t \in ]t_0 - \varepsilon_n, t_0 + \varepsilon_n[$ then

$$ \|\mathcal{T}^{-1} \circ \mathcal{V}_{t_p}[v] - \mathcal{T}^{-1} \circ \mathcal{V}_t[v]\|_{W_n^\infty(\mathbb{R})} \underset{p \to \infty}{\longrightarrow} 0. $$

**Proof** The proof is based on the fact that there exists a neighborhood $U_n$ of $t_0$ such that for all $v \in \overline{\mathcal{B}_n(0, R_n)}$, $t \in U_n \mapsto \mathcal{V}_t[v] \in W_n^\infty(\mathbb{R})$ is continuous. Since $\mathcal{T}^{-1}$ is also continuous in $W_n^\infty(\mathbb{R})$-norm, we can conclude. $\square$

**Corollary 6.8 (Continuity of the fixed-point)** Let $n \in \mathbb{N}$, for all $t, t' \in ]t_0 - \varepsilon_n, t_0 + \varepsilon_n[$,

$$ \|u_t - u_{t'}\|_{W_n^\infty(\mathbb{R})} \underset{t \to t'}{\longrightarrow} 0. $$

**Proof** Let $(t_n)_n \in (]t_0 - \varepsilon, t_0 + \varepsilon[)^N$ such that $t_n \underset{n \to \infty}{\longrightarrow} t \in ]t_0 - \varepsilon_n, t_0 + \varepsilon_n[$. First

$$ \|u_{t_n} - \mathcal{T}^{-1} \circ \mathcal{V}_{t_n}[u_t]\|_{W_n^\infty(\mathbb{R})} = \|\mathcal{T}^{-1} \circ \mathcal{V}_{t_n}[u_{t_n}] - \mathcal{T}^{-1} \circ \mathcal{V}_{t_n}[u_t]\|_{W_n^\infty(\mathbb{R})} \le k_i \|u_{t_n} - u_t\|_{W_n^\infty(\mathbb{R})} $$

and by the triangle inequality, we obtain:

$$ \|u_t - \mathcal{T}^{-1} \circ \mathcal{V}_{t_n}[u_t]\|_{W_n^\infty(\mathbb{R})} \ge \|u_{t_n} - u_t\|_{\infty} - \|u_{t_n} - \mathcal{T}^{-1} \circ \mathcal{V}_{t_n}[u_t]\|_{W_n^\infty(\mathbb{R})} \ge (1-k_n) \|u_{t_n} - u_t\|_{W_n^\infty(\mathbb{R})}. $$

Nevertheless by Lemma 6.7, $\|u_t - \mathcal{T}^{-1} \circ \mathcal{V}_{t_n}[u_t]\|_{W_n^\infty(\mathbb{R})} \underset{n \to \infty}{\longrightarrow} 0$ because $u_t$ is the fixed point of $\mathcal{T}^{-1} \circ \mathcal{V}_t$ which is a continuous operator with respect to $t$ hence $\|u_{t_n} - u_t\|_{W_n^\infty(\mathbb{R})} \underset{n \to \infty}{\longrightarrow} 0$. $\square$

**Theorem 6.9** Let $t, t_0 \in [0, 1]$,

$$ \|\rho_{V_t} - \rho_{V_{\phi,t_0}}\|_{W_n^\infty(\mathbb{R})} \underset{t \to t_0}{\longrightarrow} 0. $$

Furthermore, for all $x \in \mathbb{R}$, $k \in \mathbb{N}$, $t \mapsto \partial_x^k \rho_{V_t}(x) \in C^\infty(\mathbb{R})$ and satisfies the following partial differential equation:

$$ \partial_t \partial_x^k \rho_{V_t}(x) = \partial_x^k \left[ \left( -\phi + \int_{\mathbb{R}} \phi(s) \rho_{V_t}(s) ds \right) \rho_{V_t} \right] (x). $$

**Proof** By setting $u_t := \frac{\rho_{V_t} - \rho_{V_{t'}}}{t - t'} \frac{1}{\rho_{V_{t'}}}$,

$$ \|\rho_{V_t} - \rho_{V_{t'}}\|_{W_n^\infty(\mathbb{R})} = |t - t'| \|\rho_{V_{t'}} u_t\|_{W_n^\infty(\mathbb{R})} \le 2^n |t - t'| \|\rho_{V_{t'}}\|_{W_n^\infty(\mathbb{R})} \cdot \|u_t\|_{W_n^\infty(\mathbb{R})}. $$

By Corollary 6.8, $\|u_t\|_{W_n^\infty(\mathbb{R})} \underset{t \to t'}{\longrightarrow} \|u_{t'}\|_{W_n^\infty(\mathbb{R})}$, thus the right-hand side goes to zero proving the claim.

For the second point, we notice that

$$ \left\| \frac{\rho_{V_t} - \rho_{V_{\phi,t_0}}}{\delta t} - \rho_{V_{\phi,t_0}} u_{t_0} \right\|_{W_n^\infty(\mathbb{R})} = \left\| (u_t - u_{t_0}) \rho_{V_{\phi,t_0}} \right\|_{W_n^\infty(\mathbb{R})} \le 2^n \|u_t - u_{t_0}\|_{W_n^\infty(\mathbb{R})} \|\rho_{V_{\phi,t_0}}\|_{W_n^\infty(\mathbb{R})}. $$
Since the RHS goes to zero as $t \to t_0$, $u_{t_0} = -\phi + \int_{\mathbb{R}} \phi d\mu_{V_\phi, t_0}$ and $n$ is arbitrary, we conclude that, $x \in \mathbb{R}$, $t \mapsto \rho_{V_t}(x)$ is differentiable at every $t \in [0, 1]$ of derivative

$$ \partial_t \rho_{V_t}^{(k)}(x) = -(\rho_{V_t} \phi)^{(k)}(x) + \rho_{V_t}^{(k)}(x) \int_{\mathbb{R}} \phi d\mu_{V_t}(x). $$

Since the above expression is again differentiable in $t$ (one deals with the integral by dominated convergence theorem with the domination $|\phi(x)\partial_t \rho_{V_t}(x)| \le 2\|\phi\|_{\infty}^2(1 + \max_{s \in [0,1]} \|u_s\|_{\infty})\rho_{V_\phi, t_0}(x)$ for an arbitrary $t_0$), we conclude that for all $x \in \mathbb{R}$, $t \mapsto \rho_{V_t}(x) \in C^\infty(\mathbb{R})$. $\square$

**Corollary 6.10 (Convergence of moments)** Let $h \in \mathbb{N}$, $t \in [0, 1]$ by denoting $m(t, h) := \int_{\mathbb{R}} x^h d\mu_{V_t}$, we have for all $t_0 \in [0, 1]$

$$ m(t, h) \xrightarrow{t \to t_0} m(t_0, h). $$

**Proof** For all $t \in [0, 1]$, for all $x \in \mathbb{R}$, $\rho_{V_t}(x) \le (1 + \max_{s \in [0,1]} \|u_s\|_{\infty})\rho_{V_\phi, t_0}(x)$, hence by dominated convergence theorem $|m(t, h) - m(t_0, h)| \xrightarrow{t \to t'} 0$. $\square$

# 7 Proof of Theorem 1.4

## 7.1 Asymptotic expansion of the partition function for the Gaussian potential

The asymptotic behaviour of $\mathcal{Z}_N[V_G]$ can be deduced from Mehta's formula [Meh04, 17.6.7]

$$ \mathcal{Z}_N [V_G] = (2\pi)^{N/2} \prod_{a=1}^{N} \frac{\Gamma\left(1 + \frac{aP}{N}\right)}{\Gamma\left(1 + \frac{P}{N}\right)}. \qquad (78) $$

This will allow us to use this formula in our interpolating integration formula to deduce the asymptotic expansion of $\log \mathcal{Z}_N[V_\phi]$. From the previous equation, we can deduce the asymptotic behaviour of $\log \mathcal{Z}_N[V_G]$. It is given by the following theorem:

**Theorem 7.1** There exists a sequence $(g_k)_{k \ge 0} \in \mathbb{R}^N$, such that for all $K \ge 0$,

$$ \frac{1}{N} \log \mathcal{Z}_N [V_G] = \sum_{k=0}^{K} \frac{g_k}{N^k} + O(N^{-(K+1)}) \qquad (79) $$

with

$$ g_1 := \gamma \frac{P}{2} + \frac{\log(1 + P)}{2} + \frac{1}{2} \sum_{j \ge 1} \left( \log \left( 1 + \frac{P+1}{j} \right) - \log \left( 1 + \frac{1}{j} \right) - \frac{P}{j} \right). $$

Above $\gamma$ denotes the Euler-Mascheroni constant.

**Proof** We first use (78) to deduce

$$ \log \mathcal{Z}_N [V_G] = \frac{N \log(2\pi)}{2} + \sum_{a=1}^{N} \Gamma \left( 1 + \frac{aP}{N} \right) - N \log \Gamma \left( 1 + \frac{P}{N} \right). \qquad (80) $$
Let $K > 0$, using the Taylor series expansion of $\log \Gamma$ around $1$ (see [GR14, 8.342]), one has

$$-N \log \Gamma \left(1 + \frac{P}{N}\right) = \gamma P - \sum_{k=1}^{+\infty} \frac{\zeta(k+1)}{k+1} \frac{(-P)^{k+1}}{N^k} \quad (81)$$

where $\zeta$ denotes the Riemann $\zeta$ function. The second term in (80) can be estimated by using the Weierstrass product formula for $\frac{1}{\Gamma}$:

$$\frac{1}{\Gamma(z)} = e^{\gamma z} z \prod_{j=1}^{+\infty} \left(1 + \frac{z}{j}\right) e^{-z/j} \quad (82)$$

which is valid for any $z \in \mathbb{C}$. Hence we deduce that:

$$\begin{aligned} \sum_{k=1}^{N} \log \Gamma \left(1 + \frac{kP}{N}\right) &= - \sum_{k=1}^{N} \left( \gamma \left(1 + \frac{kP}{N}\right) + \log \left(1 + \frac{kP}{N}\right) - S_N(k) \right) \\ &= -\gamma N - \gamma \frac{(N+1)P}{2} - \sum_{k=1}^{N} \log \left(1 + \frac{kP}{N}\right) + \sum_{k=1}^{N} S_N(k) \quad (83) \end{aligned}$$

where $S_N(k) := - \sum_{j=1}^{+\infty} \left[ \log \left(1 + \frac{1}{j} + \frac{kP}{Nj}\right) - \frac{1}{j} - \frac{kP}{Nj} \right]$. By the Euler-Maclaurin formula, we have the following identity for any $K > 0$:

$$\sum_{k=1}^{N} \log \left(1 + \frac{kP}{N}\right) = \int_{0}^{N} f_{N}(t) dt + \sum_{k=1}^{K+2} \frac{B_{k}}{k!} \left(f_{N}^{(k-1)}(N) - f_{N}^{(k-1)}(0)\right) + R_{K+2}^{(N)} \quad (84)$$

where $f_N(x) := \log \left(1 + \frac{xP}{N}\right)$ and $B_k$ is the k-th Bernoulli number. The remainder $R_{K+2}^{(N)}$ is defined by $R_{K+2}^{(N)} := (-1)^{K+1} \int_0^N f_N^{(K+2)}(t) \frac{\tilde{B}_{K+2}(t - |t|)}{(K+2)!} dt$, where $\tilde{B}_{K+2}$ is the $(K+2)$-th Bernoulli polynomial. By using the following bound on Bernoulli polynomials,

$$\forall x \in [0, 1], \forall k > 0, \quad |\tilde{B}_k(x)| \le 2 \frac{k!}{(2\pi)^k} \zeta(k)$$

where $\zeta$ is the Riemann zeta function, $R_{K+2}^{(N)}$ can be controlled by the following inequalities:

$$\begin{aligned} |R_{K+2}^{(N)}| &\le \frac{2\zeta(K+2)}{(2\pi)^{K+2}} \int_0^N |f_N^{(K+2)}(t)| dt = \frac{2\zeta(K+2)}{(2\pi)^{K+2}} \int_0^N \frac{P^{K+2}}{N^{K+2}} \frac{(K+1)!}{\left(1 + \frac{P}{N}t\right)^{K+2}} dt \\ &= \frac{2\zeta(K+2)(K+1)!}{(2\pi)^{K+2}} \frac{P^{K+1}}{N^{K+1}} \int_0^P \frac{du}{(1+u)^{K+2}} = O(N^{-(K+1)}). \end{aligned}$$
Extracting the large N-behaviour in (84) leads to

$$
\begin{align}
\sum_{k=1}^{N} \log \left(1 + \frac{kP}{N}\right) &= \frac{N}{P} \int_{0}^{P} \log(1+t)dt + B_{1} \log(1+P) \nonumber \\
&\quad + \sum_{k=2}^{K+2} \frac{B_{k}}{k!} (-1)^{k} (k-2)! \frac{P^{k-1}}{N^{k-1}} \left(\frac{1}{(1+P)^{k-1}} - 1\right) + O(N^{-(K+1)}) \nonumber \\
&= Nc_{-1} + \sum_{k=0}^{K} \frac{c_{k}}{N^{k}} + O(N^{-(K+1)}) \tag{85}
\end{align}
$$

where $c_{-1} := (1 + P^{-1}) \log(1 + P) - 1$, $c_0 := \frac{\log(1 + P)}{2}$ and for all $k \in [1, K]$,

$$
c_k = \frac{-B_{k+1}(-P)^k}{k(k+1)} \left( \frac{1}{(1+P)^k} - 1 \right).
$$

Also by Fubini’s theorem, we get,

$$
\begin{align*}
\sum_{k=1}^{N} S_N(k) &= -\sum_{j=1}^{+\infty} \left\{ \sum_{k=1}^{N} \log \left( 1 + \frac{1}{j} + \frac{kP}{Nj} \right) - \frac{1}{j} - \frac{kP}{Nj} \right\} \\
&= \sum_{j=1}^{+\infty} \left\{ \frac{N}{j} + \frac{(N+1)P}{2j} + \sum_{k=1}^{N} g_{N,j}(k) \right\}
\end{align*}
$$

where $g_{N,j}(x) = \log \left(1 + \frac{1}{j} + \frac{P}{jN}x\right)$. The first equality clearly shows that the RHS is a serie of general term bounded by $O(j^{-2})$, so it converges and justifies the application of Fubini's theorem. Again by Euler-Maclaurin formula, we get:

$$
\begin{equation}
\begin{split}
\sum_{k=1}^{N} g_{N,j}(k) ={}& - \int_{0}^{N} \log \left(1 + \frac{1}{j} + t \frac{P}{Nj}\right) dt - \frac{1}{2} \left[\log\left(1 + \frac{P+1}{j}\right) - \log\left(1 + \frac{1}{j}\right)\right] \\
& - \sum_{k=2}^{K+2} \frac{(-1)^k B_k}{k(k-1)} \left\{ \left(1 + \frac{P+1}{j}\right)^{1-k} - \left(1 + \frac{1}{j}\right)^{1-k} \right\} \left(\frac{P}{Nj}\right)^{k-1} + R_{K+2}^{(N)}(j)
\end{split}
\tag{86}
\end{equation}
$$

where again the new remainder $R_{K+2}^{(N)}(j)$ can be controlled *via*

$$
\begin{align*}
|R_{K+2}^{(N)}(j)| &\le \frac{2\zeta(K+2)}{(2\pi)^{K+2}} \int_0^N \frac{(K+1)! \left(\frac{P}{Nj}\right)^{K+2}}{\left(1+\frac{1}{j}+t\frac{P}{jN}\right)^{K+2}} dt \\
&= \frac{2\zeta(K+2)}{(2\pi)^{K+2}} \left(\frac{P}{Nj}\right)^{K+1} \left\{ \left(1+\frac{P+1}{j}\right)^{-(K+1)} - \left(1+\frac{1}{j}\right)^{-(K+1)} \right\} = O\left(\frac{1}{(Nj)^{K+1}}\right)
\end{align*}
$$

where $O\left(\frac{1}{j^{K+1}}\right)$ depends on $K$ and $P$ but not on $N$. Hence we deduce that
$$ \sum_{k=1}^{N} g_{N,j}(k) = -N \int_{0}^{1} \log \left(1 + \frac{1+sP}{j}\right) ds - \frac{1}{2} \left[ \log \left(1 + \frac{P+1}{j}\right) - \log \left(1 + \frac{1}{j}\right) \right] \\ + \sum_{k=1}^{K} \frac{(-P)^k B_{k+1}}{k(k+1)} \left\{ \left(1 + \frac{P+1}{j}\right)^{-k} - \left(1 + \frac{1}{j}\right)^{-k} \right\} \frac{1}{(Nj)^k} + O \left( (jN)^{-(K+1)} \right). $$

This leads to:

$$ \sum_{k=1}^{N} S_N(k) = \sum_{j=1}^{+\infty} \left[ \frac{N}{j} + \frac{(N+1)P}{2j} + \sum_{k=1}^{N} g_{N,j}(k) \right] = \sum_{j=1}^{+\infty} \left( u_j^{(1)} N + \sum_{k=0}^{K} u_j^{(k)} N^{-k} \right) + O(N^{-(K+1)}) \\ = d_1 N + \sum_{k=0}^{K} d_{-k} N^{-k} + O(N^{-(K+1)}) \quad (87) $$

where for all $k = -1, 0, \dots, K$, $(u_j^{(k)})_{j>0} \in \ell^1(\mathbb{N}^*)$ and $d_{-k} \in \mathbb{R}$. This establishes the existence of the asymptotic expansion of $\log \mathcal{Z}_N[V_G]$ up to $O(N^{-(K+1)})$. Collecting the different terms, leads to the formula for $g_1$. $\square$

## 7.2 Free energy of the model

Only, in this subsection, since the parameter $P$ varies, we include the $P$-dependance of $\mathcal{Z}_N[V]$ in the notation and write $\mathcal{Z}_N^P[V]$ instead.

**Theorem 7.2 (Free energy formula for Gaussian Potential)** Let $P > 0$, the free energy associated with the Gaussian potential is

$$ F(P) := \lim_{N \to \infty} N^{-1} \log \mathcal{Z}_N^P [2PV_G] = -\frac{1+P}{2} \log(2P) + \frac{\log(2\pi)}{2} + \int_0^1 \log \Gamma (1+Px) dx \quad (88) $$

As $P$ goes to $+\infty$, we have:

$$ F(P) = -P \left( \frac{3 + \log 2}{2} \right) - \frac{1 + \log 2}{2} + \log(2\pi) + \frac{\log P}{12P} + O(P^{-1}) \quad (89) $$

**Proof** By a change of variable, it holds that $\mathcal{Z}_N^P[2PV_G] = (\sqrt{2P})^{-N-P(N-1)} \mathcal{Z}_N^P[V_G]$. Hence by Mehta's formula (78),

$$ F(P) := \lim_{N \to \infty} N^{-1} \log \mathcal{Z}_N^P [2PV_G] = -\frac{1+P}{2} \log(2P) + \frac{\log(2\pi)}{2} + \int_0^1 \log \Gamma (1+Px) dx. $$

We can replace the last term by its asymptotic expansion so that

$$ \int_0^1 \log \Gamma (1+Px) dx = \frac{(P+1)}{2} \log P - \frac{3P}{2} + \frac{\log(2\pi) - 1}{2} + \frac{1}{12P} \log P + O(P^{-1}). $$

We used the classic formula to conclude

$$ \log \Gamma(1+Px) = (1+Px) \log(1+Px) - 1 - Px - \frac{\log(1+Px) - \log(2\pi)}{2} + \frac{1}{12(1+Px)} + O\left(\frac{1}{(1+Px)^3}\right). $$
## 7.3 Interpolation with general potential

We first establish the link between the 1-linear statistics and the partition function with general potential and the one with Gaussian potential.

**Lemma 7.3** Let $V_t(x) = tV(x) + (1-t)V_G(x)$ with $t \in [0, 1]$. We have

$$ \log \frac{\mathcal{Z}_N[V]}{\mathcal{Z}_N[V_G]} = -N \int_0^1 \langle V - V_G \rangle_{L_N}^{V_t} dt \quad (90) $$

**Proof** By the fundamental theorem of calculus:

$$ \log \frac{\mathcal{Z}_N[V]}{\mathcal{Z}_N[V_G]} = \int_0^1 \partial_t \log \mathcal{Z}_N[V_t] dt = - \int_0^1 dt \int_{\mathbb{R}^N} p_N^{V_t}(\underline{x}) \sum_{i=1}^N \partial_t V_t(x_i) d^N \underline{x}. $$

Since $\sum_{i=1}^N \partial_t V_t(x_i) = N \int_{\mathbb{R}} [V(x) - V_G(x)] dL_N(x)$, where $L_N$ is the empirical measure associated to the external potential $V_t$, it concludes the proof. □

**Theorem 7.4** For all $\phi \in \cap_{k \ge 0} H^k(\mathbb{R})$, there exists a sequence $(c_i)_{i \ge 0} \in \mathbb{R}^N$ depending on $\phi$ and $P$ such that for all $K \ge 0$

$$ \frac{1}{N} \log \mathcal{Z}_N [V_{G,\phi}] = \sum_{i=0}^{K} \frac{c_i}{N^i} + O(N^{-(K+1)}). $$

The leading term $c_0$ is equal to the following expression:

$$ \int_{\mathbb{R}} V_{G,\phi}(x) d\mu_{V_{G,\phi}}(x) - P \iint_{\mathbb{R}^2} \log |x-y| d\mu_{V_{G,\phi}}(x) d\mu_{V_{G,\phi}}(y) + \int_{\mathbb{R}} \log \left( \frac{d\mu_{V_{G,\phi}}(x)}{dx} \right) d\mu_{V_{G,\phi}}(x). $$

The subleading term $c_1$ can be written as

$$ c_1 := \gamma \frac{P}{2} + \frac{\log(1+P)}{2} + \frac{1}{2} \sum_{j \ge 1} \left( \log \left( 1 + \frac{P+1}{j} \right) - \log \left( 1 + \frac{1}{j} \right) - \frac{P}{j} \right) \\ - P \int_0^1 \left[ \left\langle \partial_1 \Xi^{-1} \phi \right\rangle_{\mu_{V_{G,\phi,t}}} + \left\langle \Theta^{(2)} \circ \Xi_1^{-1} \left[ \partial_2 \mathcal{D} \circ \Xi^{-1} \phi \right] \right\rangle_{\mu_{V_{G,\phi,t}}} \right] dt. \quad (91) $$

**Proof** By Lemma 7.3 and Theorem 7.1, to establish the asymptotic expansion of $\mathcal{Z}_N[V_{G,\phi}]$, it suffices to obtain the one for $\int_0^1 \langle \phi \rangle_{L_N}^{V_{G,\phi,t}} dt$. By Theorem 5.2, we get

$$ \int_0^1 \langle \phi \rangle_{L_N}^{V_{G,\phi,t}} dt = \int_0^1 \langle \phi \rangle_{\mu_{V_{G,\phi,t}}} dt + \sum_{a=1}^K \frac{\int_0^1 d_a^{(1),V_{G,\phi,t}}(\phi) dt}{N^a} + \int_0^1 \left( \langle \phi \rangle_{\mathcal{L}_N}^{V_{G,\phi,t}} - \sum_{a=1}^K \frac{d_a^{(1),V_{G,\phi,t}}(\phi)}{N^a} \right) dt. $$

Finally, we conclude that the last integral is a $O(N^{-(K+1)})$ by (67) and the continuity on $t \mapsto P_{K,k}^{V_t}$ obtained in Proposition B.11. Furthermore, by collecting order 1 for $\log \mathcal{Z}_N[V_G]$ and $\int_0^1 d_a^{(1),V_{G,\phi,t}}(\phi) dt$ for $a=1$ in Theorem 5.2, we infer on the value of $c_1$. □
# 8 Conclusion

This work adapted the analysis of the loop equations method to prove the existence of a $N^{-1}$ asymptotic expansion for a general class of potential. This class include all potentials given by $x^2+\phi$ where $\phi$ is a smooth bounded function. An immediate continuation of this result would be to extend it to more general confining potentials like $x^4$ for example. Our method relied on new controls on the equilibrium measures resulting from an energy minimization and entropy maximization. A natural question would be to extend these ideas to more general interactions.

# A Appendix: Lemmas and technical results

## Lemma A.1 (Properties of the Hilbert transform)

i) As a consequence, $\pi^{-1}\mathcal{H}$ is an isometry of $L^2(\mathbb{R})$, and $\mathcal{H}$ satisfies on $L^2(\mathbb{R})$ the identity $\mathcal{H}^2 = -\pi^2 I$.

ii) Derivative: For any $f \in H^1(\mathbb{R})$, $\mathcal{H}[f]$ is also $H^1(\mathbb{R})$ and $\mathcal{H}[f]' = \mathcal{H}[f]$.

iii) For all $p > 1$, the Hilbert transform can be extended as a bounded operator $\mathcal{H}: L^p(\mathbb{R}) \to L^p(\mathbb{R})$.

iv) Skew-self adjointness: For any $f, g \in L^2(\mathbb{R})$, $\langle \mathcal{H}[f], g \rangle_{L^2(\mathbb{R})} = -\langle f, \mathcal{H}[g] \rangle_{L^2(\mathbb{R})}$.

v) For all $\delta > 0$, for all $f \in L^1(\mathbb{R})$ such that $f' \in L^\infty(\mathbb{R})$, $\|\mathcal{H}[f]\|_\infty \le (\delta^{-1}\|f\|_1 + 2\delta\|f'\|_\infty)$

**Proof** We refer to [Kin09] for the proofs of properties i)-iv). To prove v), let $f$ be such a function,

$$|\mathcal{H}[f](x)| \le \lim_{\epsilon \to 0} \left| \int_{\epsilon \le |x-y| \le \delta} \frac{f(y)dy}{y-x} \right| + \lim_{\epsilon \to 0} \left| \int_{\delta \le |x-y| \le \epsilon^{-1}} \frac{f(y)dy}{y-x} \right|.$$

The second term in the RHS can be bounded by $\delta^{-1}\|f\|_{L^1(\mathbb{R})}$ while the first term verifies

$$\lim_{\epsilon \to 0} \left| \int_{\epsilon \le |x-y| \le \delta} \frac{f(y)dy}{y-x} \right| \le \lim_{\epsilon \to 0} \int_{\epsilon \le |x-y| \le \delta} \left| \frac{f(y)-f(x)}{y-x} \right| dy + \lim_{\epsilon \to 0} \left| \int_{\epsilon \le |x-y| \le \delta} \frac{dy}{y-x} \right|.$$

The first term in the RHS can be bounded by $2\delta\|f'\|_\infty$ while the second is equal to 0. This allows to conclude. □

We recall some results obtained in [DGM23].

**Lemma A.2** Let $u \in L^2(\mathbb{R})$ be such that $\int_{\mathbb{R}} u(t)dt$ exists and let $f: t \mapsto tu(t) \in H^1(\mathbb{R})$ then

$$\mathcal{H}[u](x) \underset{|x| \to \infty}{\sim} \frac{-\int_{\mathbb{R}} u(t)dt}{x}.$$

Moreover if $\int_{\mathbb{R}} u(t)dt = 0$, $\int_{\mathbb{R}} f(t)dt$ exists and $g: t \mapsto t^2u(t) \in H^1(\mathbb{R})$, then

$$\mathcal{H}[u](x) \underset{|x| \to \infty}{\sim} \frac{-\int_{\mathbb{R}} tu(t)dt}{x^2}.$$

As a consequence, we obtain that $\mathcal{H}[\rho_V](x) \underset{|x| \to \infty}{\sim} -x^{-1}$ and the logarithmic potential $U^{\rho V}$ is Lipschitz bounded, with bounded derivative $\mathcal{H}[\rho_V]$.
**Lemma A.3** Let $n \ge 1$, and $h \in H^n(\mathbb{R})$,

$$ \|\rho_V \widetilde{\Xi}^{-1}[h]\|_{H^n(\mathbb{R})} \le C_3(V, n) \|h\|_{H^n(\mathbb{R})}. $$

with a constant $C_3(V, n)$, only depending on $V$ and $n$. For the choice of potential $V = V_{\phi,t}$, for $\phi \in C^{\infty}(\mathbb{R})$ with $\phi^{(k)} \in L^2(\mathbb{R})$ for all $k \in \mathbb{N}$ and $t \in [0, 1]$, $t \mapsto C_3(V_{\phi,t}, n)$ is a continuous function. Moreover, for all $h \in H^n(\mathbb{R}) \cap W_n^{\infty}(\mathbb{R})$,

$$ \|\rho_V \widetilde{\Xi}^{-1}[h]\|_{H^n(\mathbb{R})} \le C_5(V, n) \|h\|_{W_n^{\infty}(\mathbb{R})} $$

with a constant $C_5(V, n)$, only depending on $V$ and $n$. The function $t \mapsto C_5(V_{\phi,t}, n)$ is also continuous.

**Proof** We first prove that for all $k \ge 0$, $h \in \frac{1}{\rho_V}H^k(\mathbb{R})$, there exists finite sets of indices $\mathcal{I}_{l,a}^k$, $\mathcal{I}_l^k$ and $\mathcal{R}_l^k$ independent of $V$ and polynomials $p_{a,1,1}^k, p_{a,b,c,d}^k, q_{a,b,c}^k$ in $\theta, \dots, \theta^{(k-1)}$, with coefficients independent of $V$ and of degree at most $k$, such that for all $x \in \mathbb{R}$,

$$ \begin{aligned} \widetilde{\Xi}^{-1}[h]^{(k)} = & \sum_{a=0}^{k-1} p_{a,1,1}^k h^{(a)} + \sum_{a=0}^{k-1} \sum_{b \in \mathcal{I}_{2,a}^k} p_{a,b,2,1}^k \mathcal{H}[\rho_V p_{a,b,2,2}^k h^{(a)}] + \dots \\ & + \sum_{a=0}^{k-1} \sum_{b \in \mathcal{I}_{k,a}^k} p_{a,b,k,1}^k \mathcal{H}[\rho_V p_{a,b,k,2}^k \mathcal{H}[\dots \mathcal{H}[\rho_V p_{a,b,k,k}^k h^{(a)}]] \dots] \underbrace{+ q_{1,1}^k \widetilde{\Xi}^{-1}[h]}_{k-1} \\ & + \sum_{b \in \mathcal{I}_2^k} q_{b,2,1}^k \mathcal{H}[\rho_V q_{b,2,2}^k \widetilde{\Xi}^{-1}[h]] + \dots + \sum_{b \in \mathcal{I}_{k+1}^k} q_{b,k+1,1}^k \mathcal{H}[\rho_V q_{b,k+1,2}^k \mathcal{H}[\dots \mathcal{H}[\rho_V q_{b,k+1,k+1}^k \widetilde{\Xi}^{-1}[h]] \dots] \underbrace{+ \dots}_{k} \\ & + \left( r_{1,1}^k + \sum_{b \in \mathcal{R}_2^k} r_{b,2,1}^k \mathcal{H}[\rho_V r_{b,2,2}^k] + \dots + \sum_{b \in \mathcal{R}_k^k} r_{b,k,1}^k \mathcal{H}[\rho_V r_{b,k,2}^k \mathcal{H}[\dots \mathcal{H}[\rho_V r_{b,k,k}^k] \dots]] \underbrace{+ \dots}_{k-1} \right) \\ & \times \left( 2P \int_{\mathbb{R}} \mathcal{H}[\rho_V \widetilde{\Xi}^{-1}[h]] d\mu_V - \int_{\mathbb{R}} hd\mu_V \right). \end{aligned} \quad (92) $$

We prove it by induction, where for $n=1$ one just uses the definition of $\Xi$ for the initial case $ie$

$$ (\widetilde{\Xi}^{-1}[h])' = h - \int_{\mathbb{R}} hd\mu_V - \frac{\rho_V'}{\rho_V} (\widetilde{\Xi}^{-1}[h])' - 2P\mathcal{H}[\rho_V \widetilde{\Xi}^{-1}[h]] + 2P \int_{\mathbb{R}} \mathcal{H}[\rho_V \widetilde{\Xi}^{-1}[h]] d\mu_V. $$

For the induction step, use a bootstrap argument. Suppose (92) holds at rank $k$, then differentiate and replace $(\widetilde{\Xi}^{-1}[h])'$ by the RHS of the above relation to show that (92) holds at rank $k+1$. Now, by the Leibniz formula, for all $k \in [0, n]$, it holds that

$$ (\rho_V \Xi^{-1}[h])^{(k)} = \sum_{i=0}^{k} \binom{k}{i} \rho_V^{(k-i)} \Xi^{-1}[h]^{(i)}. $$
Furthermore by (92), by using successively that $\pi^{-1}\mathcal{H}$ is an isometry of $L^2(\mathbb{R})$, inequality (26) and Jensen's inequality, we obtain:

$$ \max_{0 \le k \le n} \| (\rho_V \Xi^{-1}[h])^{(k)}(x) \|_{L^2(\mathbb{R})} \le C_3(V, n) \|h\|_{H^k(\mathbb{R})} $$

with $C_3(V, n)$ given by

$$ C_3(V, n) := C(n) \\ \times \max_{i \le k \le n} \left\{ \begin{aligned} & i \max_{0 \le a < i} \left\| \rho_V^{(k-i)} p_{a,b,1,1}^i \right\|_{\infty} + i\pi \max_{0 \le a < i} |\mathfrak{J}_{2,a}^i| \max_{b \in \mathfrak{J}_{2,a}^i} \left( \left\| \rho_V^{(k-i)} p_{a,b,2,1}^i \right\|_{\infty} \| \rho_V p_{a,b,2,2}^i \|_{\infty} \right) + \dots \\ & + i\pi^{i-1} \max_{0 \le a < i} |\mathfrak{J}_{i,a}^i| \max_{b \in \mathfrak{J}_{i,a}^i} \left( \left\| \rho_V^{(k-i)} p_{a,b,i,1}^i \right\|_{\infty} \prod_{l=2}^i \| \rho_V p_{a,b,i,l}^i \|_{\infty} \right) + C_L \left\| \rho_V^{(k-i)} q_{1,1}^i \right\|_{\infty} \| \rho_V^{1/2} \|_{\infty} \\ & + C_L \pi |\mathfrak{J}_2^i| \max_{b \in \mathfrak{J}_2^i} \left\| \rho_V^{(k-i)} q_{b,2,1}^i \right\|_{\infty} \| \rho_V^{1/2} q_{b,2,2}^i \|_{\infty} \| \rho_V^{1/2} \|_{\infty} + \dots \\ & + C_L \pi^i |\mathfrak{J}_{i+1}^i| \max_{b \in \mathfrak{J}_{i+1}^i} \left\| \rho_V^{(k-i)} q_{b,i+1,1}^i \right\|_{\infty} \prod_{l=2}^i \| \rho_V q_{b,i+1,l}^i \|_{\infty} \| \rho_V^{1/2} q_{b,i+1,i+1}^i \|_{\infty} \| \rho_V^{1/2} \|_{\infty} \\ & + 2 \| \rho_V \|_{\infty} (1 + 2P\pi C_L) \left[ \| r_{1,1}^i \frac{\rho_V^{(k-i)}}{\rho_V} \sqrt{\rho_V} \|_{\infty} + \pi |\mathfrak{R}_2^i| \max_{b \in \mathfrak{R}_2^i} \| \rho_V^{(k-i)} r_{b,2,1}^i \|_{\infty} \| \sqrt{\rho_V} r_{b,2,1}^i \|_{\infty} + \dots \right. \\ & \qquad \left. + \pi^{i-1} |\mathfrak{R}_i^i| \max_{b \in \mathfrak{R}_i^i} \| \rho_V^{(k-i)} r_{b,i,1}^i \|_{\infty} \prod_{l=2}^{i-1} \| \rho_V r_{b,i,l}^i \|_{\infty} \| \sqrt{\rho_V} r_{b,i,l}^i \|_{\infty} \right] \end{aligned} \right\} (93) $$

For the second inequality, if $h \in H^n(\mathbb{R}) \cap W_n^\infty(\mathbb{R})$, we use the fact the same inequalities but we use the following integrals at the end $\int_{\mathbb{R}} (p_{a,b,i,l}^i h^{(a)} \rho_V(t))^2 dt \le \|h\|_{W_a^\infty(\mathbb{R})}^2 \int_{\mathbb{R}} (p_{a,b,i,l}^i \rho_{V_t})^2 dt$. This leads to
$$ \| \rho_V \Xi^{-1}[h] \|_{H^n(\mathbb{R})} \le C_5(V, n) \|h\|_{W_n^\infty(\mathbb{R})} $$

with $C_5(V, n)$ given by

$$ C_5(V, n) := C(n) \\ \times \max_{i \le k \le n} \left\{ \begin{aligned} & i \max_{0 \le a < i} \left\| \frac{\rho_V^{(k-i)}}{\rho_V} p_{a,b,1,1}^i \sqrt{\rho_V} \right\|_{\infty} + i\pi \max_{0 \le a < i} |\mathfrak{J}_{2,a}^i| \max_{b \in \mathfrak{J}_{2,a}^i} \left( \left\| \rho_V^{(k-i)} p_{a,b,2,1}^i \right\|_{\infty} \| \sqrt{\rho_V} p_{a,b,2,2}^i \|_{\infty} \right) + \dots \\ & + i\pi^{i-1} \max_{0 \le a < i} |\mathfrak{J}_{i,a}^i| \max_{b \in \mathfrak{J}_{i,a}^i} \left( \left\| \rho_V^{(k-i)} p_{a,i,1}^i \right\|_{\infty} \prod_{l=2}^{i-1} \| \rho_V p_{a,i,l}^i \|_{\infty} \| \sqrt{\rho_V} p_{a,i,l}^i \|_{\infty} \right) \\ & + C_L \left\| \rho_V^{(k-i)} q_{1,1}^i \right\|_{\infty} \| \rho_V \|_{\infty}^{1/2} + C_L \pi |\mathfrak{J}_2^i| \max_{b \in \mathfrak{J}_2^i} \left\| \rho_V^{(k-i)} q_{b,2,1}^i \right\|_{\infty} \| \rho_V^{1/2} q_{b,2,2}^i \|_{\infty} + \dots \\ & + C_L \pi^i |\mathfrak{J}_{i+1}^i| \max_{b \in \mathfrak{J}_{i+1}^i} \left\| \rho_V^{(k-i)} q_{b,i+1,1}^i \right\|_{\infty} \prod_{l=2}^i \| \rho_V q_{b,i+1,l}^i \|_{\infty} \| \rho_V^{1/2} q_{b,i+1,i+1}^i \|_{\infty} \\ & + 2 \left( 1 + 2P\pi \| \rho_V \|_{\infty}^{1/2} C_L \right) \left[ \| r_{1,1}^i \frac{\rho_V^{(k-i)}}{\rho_V} \sqrt{\rho_V} \|_{L^2(\mathbb{R})} + \pi |\mathfrak{R}_2^i| \max_{b \in \mathfrak{R}_2^i} \| \rho_V^{(k-i)} r_{b,2,1}^i \|_{\infty} \| \sqrt{\rho_V} r_{b,2,1}^i \|_{\infty} + \dots \right. \\ & \qquad \left. + \pi^{i-1} |\mathfrak{R}_i^i| \max_{b \in \mathfrak{R}_i^i} \| \rho_V^{(k-i)} r_{b,i,1}^i \|_{\infty} \prod_{l=2}^{i-1} \| \rho_V r_{b,i,l}^i \|_{\infty} \| \sqrt{\rho_V} r_{b,i,l}^i \|_{\infty} \right] \end{aligned} \right\} (94) $$
The fact that $t \mapsto C_i(V_{\phi,t}, n)$ is shown in Appendix B.

□

**Remark A.4** With $\theta = \frac{\rho'_V}{\rho_V}$, $g := \Xi^{-1}[f]$ and $c := \int_{\mathbb{R}} (2PH[\rho_V g] - f) d\mu_V$ we have

$$g' = f - \theta g - 2PH[\rho_V g] + c$$

$$g'' = -\theta f + f' - 2PH[\rho_V f] + (\theta^2 - \theta') g + 2P\theta H[\rho_V g] + 4P^2 H[\rho_V H[\rho_V g]] + (-\theta - 2PH[\rho_V]) c.$$

$$g^{(3)} = (\theta^2 - 2\theta') f - \theta f' + f'' + 2P\theta H[\rho_V f] - 2PH[\rho_V \theta f] - 2PH[\rho_V f'] + 4P^2 H[\rho_V H[\rho_V f]] \\ + [(\theta^2 - \theta')' - (\theta^3 - \theta\theta')] g + 2P(\theta^2 - 2\theta') H[\rho_V g] - 4P^2\theta H[\rho_V H[\rho_V g]] + 4P^2 H[\rho_V \theta H[\rho_V g]] \\ + 4P^2 H[\rho_V H[\rho_V \theta g]] - 8P^3 H[\rho_V H[\rho_V H[\rho_V g]]] \\ + ((\theta^2 - 2\theta') + 2P\theta H[\rho_V] - 2PH[\rho_V \theta] + 4P^2 H[\rho_V H[\rho_V]]) c$$

# B Integrability of the constants

## B.1 Parameter continuity of norms of certain functions

In this appendix, we work with $V = V_{G,\phi,t}$, $t \in [0,1]$ and $\phi \in \cap_{k \ge 0} H^k(\mathbb{R})$. We will show, that the constant $C_i(V_{G,\phi,t}, n)$ appearing in our problem, see Theorem 4.14, Theorem 4.15 and 4.18 will be continuous in $t$ hence integrable on $[0,1]$. In this section, we denote for all $t \in [0,1]$,

$$\alpha_t := \frac{\rho_{V_{G,\phi,t}}}{\rho'_{V_{G,\phi,t}}} \quad \text{and} \quad \theta_t := \frac{\rho'_{V_{G,\phi,t}}}{\rho_{V_{G,\phi,t}}}.$$

First of all, by Theorem 6.9, the map $t \mapsto \|\rho_{V_{G,\phi,t}}\|_{W_n^\infty(\mathbb{R})}$ is continuous for all $n \in \mathbb{N}$. This allows to conclude that

**Lemma B.1** Let $t, t_0 \in [0,1]$, for all $n \in \mathbb{N}$,

$$\|\mathcal{H}[\rho_{V_{G,\phi,t}} - \rho_{V_{G,\phi,t_0}}]\|_{W_n^\infty(\mathbb{R})} \xrightarrow{t \to t_0} 0.$$

**Proof** We prove it by induction and use Lemma A.1 and Theorem 6.9. For $n = 0$, we know that there exists $C > 0$, such that:

$$\|\mathcal{H}[\rho_{V_{G,\phi,t}} - \rho_{V_{G,\phi,t_0}}]\|_{\infty} \le C \left( \|\rho_{V_{G,\phi,t}} - \rho_{V_{G,\phi,t_0}}\|_{L^1(\mathbb{R})} + \|\rho'_{V_{G,\phi,t}} - \rho'_{V_{G,\phi,t_0}}\|_{\infty} \right).$$

By Scheffé's lemma, the $L^1$ norm goes to zero and by Theorem 6.9 goes also to zero as $t$ goes to $t_0$.

Now suppose that $\|\mathcal{H}[\rho_{V_{G,\phi,t}} - \rho_{V_{G,\phi,t_0}}]\|_{W_n^\infty(\mathbb{R})} \xrightarrow{t \to t_0} 0$ for some $n \ge 0$. We have that

$$\|\mathcal{H}[\rho_{V_{G,\phi,t}}^{(n+1)} - \rho_{V_{G,\phi,t_0}}^{(n+1)}]\|_{\infty} \le C \left( \|\rho_{V_{G,\phi,t}}^{(n+1)} - \rho_{V_{G,\phi,t_0}}^{(n+1)}\|_{L^1(\mathbb{R})} + \|\rho_{V_{G,\phi,t}}^{(n+2)} - \rho_{V_{G,\phi,t_0}}^{(n+2)}\|_{\infty} \right).$$

Since for all $x \in \mathbb{R}$, $\rho_{V_{G,\phi,t_0}}^{(n+1)}(x)$ goes to zero and we have the following domination by Leibniz formula

$$|\rho_{V_{G,\phi,t}}^{(n+1)}(x)| \le (1 + \max_{s \in [0,1]} \|u_s\|_{W_{n+1}^\infty(\mathbb{R})}) \sum_{k=0}^{n+1} |\rho_{V_{G,\phi,t_0}}^{(k)}(x)|.$$

By dominated convergence theorem, the $L^1$-norm goes to zero and the last term also trivially goes to zero by theorem 6.9. □
Secondly, we can set $M_{V_{G,\phi,t}}$ (see Lemma 4.9) equal to any value $M$ such that

$$M > \max \left( 1 + \|\phi'\|_{\infty} + 2P \max_{t \in [0,1]} \|\mathcal{H}[\rho_{V_{G,\phi,t}}]\|_{\infty}, 2 \left( \|\phi'\|_{\infty} + 2P \max_{t \in [0,1]} \|\mathcal{H}[\rho_{V_{G,\phi,t}}]\|_{\infty} \right) \right).$$

We choose such a $M$, it is well-defined because of Lemma B.1.

**Lemma B.2** For all $t \in [0, 1]$, for all $i, j \in \mathbb{N}$, for all $|x| \ge M$,

$$\left| \frac{\rho'_{V_{G,\phi,t}}(x)}{\rho_{V_{G,\phi,t}}(x)} \right| \ge 1 \quad \text{and} \quad \left| \left( \frac{\rho_{V_{G,\phi,t}}}{\rho'_{V_{G,\phi,t}}} \right)^{(i)} (x)^j \right| \le \delta_{i,0} \frac{C_{0,j}}{|x|^j} + \frac{C_{i,j}}{|x|^{2j}}$$

for constants $C_{i,j} > 0$ independent of $t$.

**Proof** Let $x \in \mathbb{R}$, $-\frac{\rho'_{V_{G,\phi,t}}(x)}{\rho_{V_{G,\phi,t}}(x)} = x + t\phi'(x) + 2P\mathcal{H}[\rho_{V_{G,\phi,t}}](x)$. Thus if $|x| \ge M$,

$$\left| \frac{\rho'_{V}(x)}{\rho_{V}(x)} \right| \ge 1 + \left( \|\phi'\|_{\infty} - t|\phi'(x)| \right) + 2P \left( \max_{s \in [0,1]} \|\mathcal{H}[\rho_{V_{G,\phi,s}}]\|_{\infty} - |\mathcal{H}[\rho_{V_{G,\phi,t}}](x)| \right) \ge 1.$$

For the second point, one notices by differentiation and (17) that there exists polynomials $P_k$ with coefficients independent of $t$ such that

$$\left( \frac{\rho_{V_{G,\phi,t}}}{\rho'_{V_{G,\phi,t}}} \right)^{(i)} (x) = \sum_{k=1}^{i} \frac{P_k \left( t\phi'(x), \dots, t\phi^{(i+1)}(x), \mathcal{H}[\rho_{V_{G,\phi,t}}](x), \dots, \mathcal{H}\left[\rho_{V_{G,\phi,t}}^{(i)}\right](x) \right)}{\left( x + t\phi'(x) + 2P\mathcal{H}[\rho_{V_{G,\phi,t}}](x) \right)^{k+1}}.$$

Furthermore, since $|x| \ge 2 (\|\phi'\|_{\infty} + 2P \max_{t \in [0,1]} \|\mathcal{H}[\rho_{V_{G,\phi,t}}]\|_{\infty})$ we have

$$\left| x + t\phi'(x) + 2P\mathcal{H}[\rho_{V_{G,\phi,t}}](x) \right| \ge \frac{|x|}{2} + \left( \frac{|x|}{2} - \|\phi'\|_{\infty} - 2P \max_{s \in [0,1]} \|\mathcal{H}[\rho_{V_s}]\|_{\infty} \right) \ge \frac{|x|}{2}.$$

Finally, the whole dependence in $t$ and $x$ of the numerator are in the entries which are bounded uniformly in $t$ and $x$, we can conclude that each numerator in the sum is bounded by a constant

$C_k > 0$. We can conclude that $\left| \left( \frac{\rho_{V_{G,\phi,t}}}{\rho'_{V_{G,\phi,t}}} \right)^{(i)} (x) \right| \le 4i \max_{k \le i} C_k |x|^{-2}$, raising to the power $j$ leads to the conclusion. $\square$

**Lemma B.3** The map $t \mapsto \|\rho_{V_{G,\phi,t}}^{-1}\|_{L^{\infty}([-M,M])}$ is continuous.

**Proof** Let $x \in [-M, M]$,

$$|\rho_{V_{G,\phi,t}}(x)^{-1} - \rho_{V_{G,\phi,t_0}}(x)^{-1}| = \left| \frac{\delta t u_t(x) \rho_{V_{G,\phi,t_0}}(x)}{\rho_{V_{G,\phi,t_0}}(x) \rho_{V_{G,\phi,t}}(x)} \right| \le |\delta t| \frac{\|u_t\|_{\infty} \|\rho_{V_{G,\phi,t_0}}^{-1}\|_{L^{\infty}([-M,M])}}{(1 - |\delta t| \|u_t\|_{\infty})}.$$

Taking the supremum over $x \in [-M, M]$ and let $t$ goes to $t_0$ establishes the result. $\square$

Now it remains to bound the $L^2$ or $L^{\infty}$ norms of the functions $f^{(i),V_{G,\phi,t}}$ and $\mathcal{I}_a^{V_{G,\phi,t}}$ appearing in Theorem 4.14, 4.15 and 4.18.

**Lemma B.4** $t \mapsto \|\mathcal{I}_a^{V_{G,\phi,t}}\|_{\infty}$ for all $a \in \{1, 2\}$ is continuous where $\mathcal{I}_a^{V_{G,\phi,t}}$ is defined in (52).
**Proof** Let $x > 0$, $t, t_0 \in [0, 1]$, by the mean-value theorem and with $u_t$ defined in Section 6, we get

$$
\begin{aligned}
|\mathcal{I}_1^{V_{G,\phi,t}}(x) - \mathcal{I}_1^{V_{G,\phi,t_0}}(x)| &= \left| \frac{1}{\rho_{V_{G,\phi,t}}(x)} \int_x^{+\infty} \rho_{V_{G,\phi,t}}(s)ds - \frac{1}{\rho_{V_{G,\phi,t_0}}(x)} \int_x^{+\infty} \rho_{V_{G,\phi,t_0}}(s)ds \right| \\
&\le \frac{1}{\rho_{V_{G,\phi,t_0}}(x)} \int_x^{+\infty} \rho_{V_{G,\phi,t_0}}(s)ds \left| \frac{1+\delta t u_t(s)}{1+\delta t u_t(x)} - 1 \right| \\
&= \frac{|\delta t|}{\rho_{V_{G,\phi,t_0}}(x)(1-|\delta t||u_t||_\infty)} \int_x^{+\infty} \rho_{V_{G,\phi,t_0}}(s)ds |u_t(s) - u_t(x)| \\
&\le \frac{|\delta t||u'_t||_\infty}{(1-|\delta t||u_t||_\infty)} \frac{1}{\rho_{V_{G,\phi,t_0}}(x)} \int_x^{+\infty} (s-x)\rho_{V_{G,\phi,t_0}}(s)ds. \quad (95)
\end{aligned}
$$

One thus concludes by showing that $x \in ]0, +\infty[ \mapsto \frac{1}{\rho_{V_{G,\phi,t_0}}(x)} \int_x^{+\infty} (s-x)\rho_{V_{G,\phi,t_0}}(s)ds$ is bounded, since, in (95), $\delta t$ goes to zero and $t \mapsto \|u_t\|_{W_1^\infty(\mathbb{R})}$ is bounded.

This can be proven by integration by parts

$$
\frac{1}{\rho_{V_{G,\phi,t_0}}(x)} \int_x^{+\infty} (s-x) \rho_{V_{G,\phi,t_0}}(s) ds = \frac{1}{\rho_{V_{G,\phi,t_0}}(x)} \left[ \frac{\rho_{V_{G,\phi,t_0}}(s) \rho_{V_{G,\phi,t_0}}(s) s}{\rho'_{V_{G,\phi,t_0}}(s)} \right]_x^{+\infty} \\
- \frac{1}{\rho_{V_{G,\phi,t_0}}(x)} \int_x^{+\infty} \left[ \frac{\rho_{V_{G,\phi,t_0}}(s)}{\rho'_{V_{G,\phi,t_0}}(s)} + \left( \frac{\rho_{V_{G,\phi,t_0}}(s)}{\rho'_{V_{G,\phi,t_0}}(s)} \right)' \right] \rho_{V_{G,\phi,t_0}}(s) ds.
$$

The first term in the right hand side is bounded, while by assumption v), the last term is a

$$
\frac{1}{\rho_{V_{G,\phi,t_0}}(x)} \int_x^{+\infty} O \left( \frac{\rho_{V_{G,\phi,t_0}}(s)}{s} \right) ds = O \left( \frac{1}{\rho_{V_{G,\phi,t_0}}(x)} \int_x^{+\infty} \frac{\rho_{V_{G,\phi,t_0}}(s)}{s} ds \right).
$$

Again by an integration by parts, the last integral in the remainder is equal to

$$
\frac{\rho_{V_{G,\phi,t_0}}(x)}{\rho'_{V_{G,\phi,t_0}}(x)x} + O\left(\frac{1}{\rho_{V_{G,\phi,t_0}}(x)} \int_x^{+\infty} \frac{\rho_{V_{G,\phi,t_0}}(s)}{s^2} ds\right).
$$

Since $\rho_{V_{G,\phi,t_0}}$ is decreasing in a neighborhood at infinity the last remainder is a $\mathcal{O}(1)$ while the first is behaves like $x^{-2}$ at infinity. Finally, $x \mapsto \frac{x}{\rho_{V_{G,\phi,t_0}}(x)} \int_x^{+\infty} \rho_{V_{G,\phi,t_0}}(s)ds$ is bounded by the same exact technique.

Doing the same thing over $]-\infty, 0]$ establishes that $t \mapsto \|\mathcal{I}_1^{V_{G,\phi,t}}\|_{\infty}$ is continuous.

Just as before, we get by the mean-value theorem,

$$
\begin{aligned}
|\mathcal{I}_2^{V_{G,\phi,t}}(x)^2 - \mathcal{I}_2^{V_{G,\phi,t_0}}(x)^2| &\le \frac{1}{\rho_{V_{G,\phi,t_0}^2}(x)} \int_x^{+\infty} \rho_{V_{G,\phi,t_0}^2}(s) ds \left| \frac{(1+\delta t u_t(s))^2}{(1+\delta t u_t(x))^2} - 1 \right| \\
&\le \frac{|\delta t| \|u'_t\|_{\infty} (2+\|u_t\|_{\infty})}{(1-|\delta t| \|u_t\|_{\infty})^2} \frac{1}{\rho_{V_{G,\phi,t_0}}(x)^2} \int_x^{+\infty} (s-x) \rho_{V_{G,\phi,t_0}}(s)^2 ds. \quad (96)
\end{aligned}
$$

We conclude by showing that $x \in [0, +\infty[ \mapsto \frac{1}{\rho_{V_{G,\phi,t_0}}(x)^2} \int_x^{+\infty} (s-x)\rho_{V_{G,\phi,t_0}}(s)^2 ds$ is bounded which can again be proven by the same integration by parts and by doing the exact same thing on $]-\infty, 0]$.
Therefore by the fact that

$$0 \le \left| \| \mathcal{I}_2^{V_{G,\phi,t}} \|_{\infty}^2 - \| \mathcal{I}_2^{V_{G,\phi,t_0}} \|_{\infty}^2 \right| \le \| (\mathcal{I}_2^{V_{G,\phi,t}})^2 - (\mathcal{I}_2^{V_{G,\phi,t_0}})^2 \|_{\infty} \xrightarrow{t \to t_0} 0$$

we conclude that $t \mapsto \|\mathcal{I}_2^{V_{G,\phi,t}}\|_{\infty}$ is continuous. $\square$

Next, we show that any polynomial in $\theta_t := \frac{\rho'_{V_{G,\phi,t}}}{\rho_{V_{G,\phi,t}}}$ and its derivatives yield a continuous dependence in $t$.

**Lemma B.5** Let $P$ a polynomial in $\theta_t, \dots, \theta_t^{(k)}$ for some $k \ge 0$ with coefficients independent of $t$, let $l \in \mathbb{N}$ then the two following maps are continuous:

$$(i) \quad t \mapsto \|P(\theta_t, \dots, \theta_t^{(k)})\|_{L^{\infty}([-M,M])}$$

$$(ii) \quad t \mapsto \| \sqrt{\rho_{V_{G,\phi,t}}^{(l)}} P(\theta_t, \dots, \theta_t^{(k)}) \|_{L^{\infty}(\mathbb{R})}$$

$$(iii) \quad t \mapsto \| \rho_{V_{G,\phi,t}}^{(l)} P(\theta_t, \dots, \theta_t^{(k)}) \|_{L^{\infty}(\mathbb{R})}.$$

**Proof** Proving these continuity results for any monomial in those variables is enough. Furthermore, since by continuity $x \in [-M, M] \mapsto \theta_t^{(i)}$ is bounded for all $i \le k$, thus this monomial in $(\theta_t^{(i)})_{0 \le i \le k}$ converges uniformly to the monomial in $(\theta_{t_0}^{(i)})_{0 \le i \le k}$ as $t$ goes to $t_0$. The arguments are that the product of two bounded, uniformly converging sequences of functions converges to the product of the limits and that for all $i > 0$, $\theta_t^{(i)}(x) - \theta_{t_0}^{(i)}(x) = -\delta t \phi^{(i+1)}(x) - 2PH[\rho_{V_{G,\phi,t}}^{(i)} - \rho_{V_{G,\phi,t_0}}^{(i)}]$. The latter, when taking the supremum over $x \in [-M, M]$, goes to zero by Lemma B.1. This establishes (i). Furthermore, notice that (iii) implies (ii) since $\|\sqrt{\rho_{V_{G,\phi,t}}^{(l)}} P(\theta_t, \dots, \theta_t^{(k)})\|_{L^{\infty}(\mathbb{R})} = \|\rho_{V_{G,\phi,t}}^{(l)} P(\theta_t, \dots, \theta_t^{(k)})\|_{L^{\infty}(\mathbb{R})}^{1/2}$ and $P$ is arbitrary so we only prove (iii). Moreover since by Faà di Bruno's formula $\rho_{V_{G,\phi,t}}^{(l)} = \exp(\log \rho_{V_{G,\phi,t}}^{(i)})$ can be written as $Q(\theta, \dots, \theta^{(l)}) \rho_V$ where $Q$ is a polynomial with coefficients independent of $t$, it suffices to prove the result for $l = 0$.

For all $i \in \mathbb{N}$, we have $\theta_t^{(i)}(x) = -\delta_{i,0}x - t\phi^{(i+1)}(x) - 2PH[\rho_{V_{G,\phi,t}}^{(i)}]$. Noticing that by Leibniz formula and the mean value theorem that for all $j \in \mathbb{N}$, for all $0 < \alpha < 1$

$$\begin{aligned} \left| x^j \rho_{V_{G,\phi,t}}(x)^\alpha - x^j \rho_{V_{G,\phi,t_0}}(x)^\alpha \right| & \le \left| x^j \rho_{V_{G,\phi,t_0}}(x)^\alpha \right| \cdot \left| (1 + \delta t u_t(x))^\alpha - 1 \right| \\ & \le \frac{\alpha |\delta t| \max_{s \in [0,1]} \|u_s\|_{W_l^\infty(\mathbb{R})}}{\left(1 - |\delta t| \max_{s \in [0,1]} \|u_s\|_{W_l^\infty(\mathbb{R})}\right)^{1-\alpha}} \left\| x \mapsto x^j \rho_{V_{G,\phi,t_0}}(x)^\alpha \right\|_{\infty} \end{aligned}$$

where the existence of the max is justified by Corollary 6.8. Taking the supremum over $x \in \mathbb{R}$ and let $t$ goes to $t_0$ shows that $t \mapsto (x \mapsto x^j \rho_{V_{G,\phi,t}}(x)^\alpha) \in L^\infty(\mathbb{R})$ is continuous. By boundedness and continuity with respect to the $t$ parameter of $t \mapsto \phi^{(i+1)}(x) + 2PH[\rho_{V_{G,\phi,t}}^{(i)}]$, we deduce that for all $i \in \mathbb{N}$ and $\alpha > 0$,

$$\| \theta_t^{(i)} \rho_{V_{G,\phi,t}}(x)^\alpha - \theta_{t_0}^{(i)} \rho_{V_{G,\phi,t_0}}(x)^\alpha \|_{\infty} \xrightarrow{t \to t_0} 0.$$

From this last uniform convergence result, we show that by taking a monomial $\prod_{i=0}^k (\theta_t^{(i)})^{l_i}$ such that $\sum_{i=0}^k l_i = m$, we deduce that $\rho_{V_{G,\phi,t}} \prod_{i=0}^k (\theta_t^{(i)})^{l_i} = \prod_{i=0}^k (\theta_t^{(i)} \sqrt[m]{\rho_{V_{G,\phi,t}}})^{l_i}$, as a product of
bounded, uniformly converging t-sequences of functions, it converges uniformly. This concludes the proof. □

**Lemma B.6 (Continuity of uniform norms)** For all $j \in \{1, 2, 3, 5, 6\}$, $t \mapsto \|f_{n_1, a, b}^{(j), V_{G, \phi, t}}\|_{L^\infty([-M, M]^c)}$ is continuous where we recall that

$$
\begin{align*}
f_{l, a, b}^{(1), V_{G, \phi, t}} : x &\mapsto Q_a^l \left( \theta_t, \dots, \theta_t^{(a)} \right) (x) \alpha_t(x) P_b^{l-a} \left( \alpha_t, \dots, \alpha_t^{(b)} \right) (x), \\
f_{l, a, b}^{(2), V_{G, \phi, t}} : x &\mapsto \frac{Q_a^l \left( \theta_t, \dots, \theta_t^{(a)} \right) (x)}{\rho_{V_{G, \phi, t}}(x)} \int_x^{\operatorname{sgn}(x)\infty} \left[ \alpha_t P_b^{l-a} \left( \alpha_t, \dots, \alpha_t^{(b)} \right) \right]' (y) \rho_{V_{G, \phi, t}}(y) dy, \\
f_{l, a, b}^{(3), V_{G, \phi, t}} : x &\mapsto \frac{Q_a^l \left( \theta_t, \dots, \theta_t^{(a)} \right) (x)}{\rho_{V_{G, \phi, t}}(x)} \left| \int_x^{\operatorname{sgn}(x)\infty} \alpha_t(y)^2 P_b^{l-a} \left( \alpha_t, \dots, \alpha_t^{(b)} \right) (y)^2 \rho_{V_{G, \phi, t}}(y)^2 dy \right|^{1/2}, \\
f_{l, a, b}^{(5), V_{G, \phi, t}} : x &\mapsto \frac{Q_a^l \left( \theta_t, \dots, \theta_t^{(a)} \right) (x)}{\rho_{V_{G, \phi, t}}(x)} \int_x^{\operatorname{sgn}(x)\infty} \left| P_b^{l-a} \left( \alpha_t, \dots, \alpha_t^{(b)} \right) (y) \right| \rho_{V_{G, \phi, t}}(y) dy, \\
f_{l, a, b}^{(6), V_{G, \phi, t}} : x &\mapsto \frac{\left| Q_a^l \left( \theta_t, \dots, \theta_t^{(a)} \right) (x) \right|}{\rho_{V_{G, \phi, t}}(x)} \left| \int_x^{\operatorname{sgn}(x)\infty} \left| P_b^{l-a} \left( \alpha_t, \dots, \alpha_t^{(b)} \right) (y) \right|^2 \rho_{V_{G, \phi, t}}(y)^2 dy \right|^{1/2}.
\end{align*}
$$

**Proof** First, one can check that, from Lemma 4.13, $f_{n_1, a, b}^{(1), V_{G, \phi, t}}(x) = O(x^{-1})$ for all $t \in [0, 1]$.

Noticing that since $\alpha(x) = (-x - t\phi'(x) - 2PH[\rho_{V_{G,\phi,t}}](x))^{-1} = \theta(x)^{-1}$, there exists $n > 0$ and a polynomial expression $P$ with coefficients independent of $t$ such that

$$
f_{n_1, a, b}^{(1), V_{G, \phi, t}}(x) = \frac{\mathbf{P} \left( x^{-1}, t\phi', \dots, t\phi^{(k)}, P\mathcal{H}[\rho_{V_{G, \phi, t}},] \dots, P\mathcal{H}[\rho_{V_{G, \phi, t}}^{(k-1)}] \right)}{(1 + t\phi'(x)x^{-1} + 2P\mathcal{H}[\rho_{V_{G, \phi, t}}](x)x^{-1})^n}.
$$

In the above expression, the numerator must be a $O(x^{-1})$. We conclude from this closed form, that $f_{n_1, a, b}^{(1), V_{G, \phi, t}}$ converges uniformly to $f_{n_1, a, b}^{(1), V_{G, \phi, t_0}}$ when $t$ goes to $t_0$ on $[-M, M]^c$. Indeed, $f_{n_1, a, b}^{(1), V_{G, \phi, t}}$ is a bounded rational function such that the denominator is bounded from below uniformly in $t$ (see Lemma B.2) and such that both the numerator and denominator converges uniformly. Thus, $t \mapsto \|f_{n_1, a, b}^{(1), V_{G, \phi, t}}\|_{L^\infty([-M, M]^c)}$ is continuous.

We only prove the continuity of $t \mapsto \|f_{n_1, a, b}^{(j), V_{G, \phi, t}}\|_{L^\infty([-M, M]^c)}$ in the case $j = 5$, since the same arguments also prove the cases $j \in \{2, 3, 6\}$. Since $|\rho_{V_{G, \phi, t}}(x) - \rho_{V_{G, \phi, t_0}}(x)| \le |\delta t| \max_{s \in [0, 1]} \|u_s\|_\infty |\rho_{V_{G, \phi, t_0}}(x)$ and that the following map is uniformly bounded in $t \in [0, 1]$ and $x > M$

$$
g_t : x \mapsto \frac{Q_a^l \left( \theta_t, \dots, \theta_t^{(a)} \right) (x)}{\rho_{V_{G,\phi,t}}(x)} \int_x^{+\infty} \left| P_b^{l-a} \left( \alpha_t, \dots, \alpha_t^{(b)} \right) (y) \right| \rho_{V_{G,\phi,t_0}}(y) dy,
$$

we can just show that $g_t$ converges uniformly to $t_0$ as $t$ goes to $t_0$. Moreover since

$$
|\rho_{V_{G, \phi, t}}(x)^{-1} - \rho_{V_{G, \phi, t_0}}(x)^{-1}| \le |\delta t| \frac{\max_{s \in [0,1]} \|u_s\|_\infty}{1 - |\delta t| \max_{s \in [0,1]} \|u_s\|_\infty} \rho_{V_{G, \phi, t_0}}(x)^{-1},
$$
it is enough to show the uniform convergence for $\frac{\rho_{V_{G,\phi,t}}}{\rho_{V_{G,\phi,t_0}}}g_t$. One can also notice that, for constants $C_{l,a}$ and $C_{l,a,b}$ independent of $t$,

$$|Q_a^l(\theta_t, \dots, \theta_t^{(a)})(x)| \le C_{l,a} |x|^{l-a} \quad \text{and} \quad \left|P_b^{l-a}(\alpha_t, \dots, \alpha_t^{(b)})(y)\right| \le C_{l,a,b} |x|^{-(l-a)}.$$

Finally, by writing $|x^{-(l-a)}Q_a^l(\theta_t, \dots, \theta_t^{(a)})(x)|$ as polynomial in $x^{-1}$, $t\phi^{(i+1)}(x)$ and $\mathcal{H}[\rho_{V_{G,\phi,t}}^{(i)}]$ for $i \ge 0$ and $x^{l-a}P_b^{l-a}(\alpha_t, \dots, \alpha_t^{(b)})(x)$ as a rational function in those same variables, we conclude that these functions converge uniformly towards the same functions at $t_0$. Therefore

$$x \mapsto x^{-(l-a)} Q_a^l(\theta_t, \dots, \theta_t^{(a)})(x) \frac{x^{l-a}}{\rho_{V_{G,\phi,t_0}}(x)} \int_x^{+\infty} \frac{\rho_{V_{G,\phi,t_0}}(y)dy}{y^{l-a}} |y^{l-a} P_b^{l-a}(\alpha_t, \dots, \alpha_t^{(b)})(y)|$$

converges uniformly to the same functions at $t_0$. This establishes the proposition. $\square$

**Lemma B.7 (Continuity $L^2$-norms)** For all $j \in [1, 4]$, the maps $t \mapsto \|f_{n_1,a,b}^{(j),V_{G,\phi,t}}\|_{L^2([-M,M]^c)}$ are continuous where

$$f_{n_1,a,b}^{(4),V_{G,\phi,t}} : x \mapsto \frac{Q_a^{n_1}(\theta_t, \dots, \theta_t^{(a)})(x)}{\rho_{V_{G,\phi,t}}(x)} \left| \int_x^{\operatorname{sgn}(x)\infty} \left[ \alpha_t P_b^{n_1-a}(\alpha_t, \dots, \alpha_t^{(b)}) \right]'(y) \rho_{V_{G,\phi,t}}(y)^2 dy \right|^{1/2} . \quad (97)$$

**Proof** For the continuity of $t \mapsto \|f_{n_1,a,b}^{(1),V_{G,\phi,t}}\|_{L^2([-M,M]^c)}$, we use dominated convergence theorem. Since, we showed uniform convergence and that $\|f_{n_1,a,b}^{(1),V_{G,\phi,t}}\|_{L^\infty([-M,M]^c)} < +\infty$, we conclude that for all $x \in [-M, M]^c$,

$$f_{n_1,a,b}^{(1),V_{G,\phi,t}}(x)^2 \xrightarrow[t \to t_0]{} f_{n_1,a,b}^{(1),V_{G,\phi,t_0}}(x)^2.$$

The domination follows from the fact $f_{n_1,a,b}^{(1),V_{G,\phi,t}}(x) = O(x^{-2})$ and that all the dependence in $t$ is bounded, hence there exists a constant $C_{n_1,a,b} > 0$ independent of $t$ such that, for all $x \in [-M, M]^c$ and all $t \in [0, 1]$,

$$|f_{n_1,a,b}^{(1),V_{G,\phi,t}}(x)| \le \frac{C}{x^2}.$$

This establishes that $\|f_{n_1,a,b}^{(1),V_{G,\phi,t}}\|_{L^2([-M,M]^c)} \xrightarrow[t \to t_0]{} \|f_{n_1,a,b}^{(1),V_{G,\phi,t_0}}\|_{L^2([-M,M]^c)}$.

We now establish the continuity for $t \mapsto \|f_{n_1,a,b}^{(3),V_{G,\phi,t}}\|_{L^2([-M,M]^c)}$, the case $j \in \{2, 4\}$ is done with the exact same arguments. We want to use dominated convergence theorem, for $(f_{n_1,a,b}^{(3),V_{G,\phi,t}})^2$. The latter, when $t \to t_0 \in [0, 1]$, also converges uniformly since it is uniformly bounded and that we proved that $f_{n_1,a,b}^{(3),V_{G,\phi,t}}$ converges uniformly. It just remains to verify the domination hypothesis. By Lemma 4.13, we know that $\alpha(y)^2 P_b^{n_1-a}(\alpha_t, \dots, \alpha_t^{(b)})(y)^2 = O(y^{-2(n_1-a+1)})$. We conclude by Lemma B.2 that there exists a constant $C_{n_1,a,b} > 0$ such that for all $y > M$,

$$\alpha_t(y)^2 P_b^{n_1-a}(\alpha_t, \dots, \alpha_t^{(b)})(y)^2 \le \frac{C_{n_1,a,b}}{y^{2(n_1-a+1)}}.$$
Similarly $|Q_a^{n_1}(\theta_t, \dots, \theta_t^{(a)})(y)^2| \le C_{n_1,a}|x|^{2(n_1-a)}$ for all $y > M$ and for $C_{n_1,a} > 0$ a constant independent of $t$ and $y$. Finally, we get the following domination for an arbitrary $t_0$ and all $y > M$

$$ \frac{Q_a^{n_1}(\theta_t, \dots, \theta_t^{(a)})(x)^2}{\rho_{V_{G,\phi,t}}(x)^2} \int_x^{+\infty} \alpha_t(y)^2 P_b^{n_1-a}(\alpha_t, \dots, \alpha_t^{(b)})(y)^2 \rho_{V_{G,\phi,t}}(y)^2 dy \\ \le \frac{C_{n_1,a}|x|^{2(n_1-a)}}{\rho_{V_{G,\phi,t_0}}(x)(1-\max_{s \in [0,1]} \|u_s\|_{\infty})} \int_x^{+\infty} \frac{C_{n_1,a,b}(1+\max_{s \in [0,1]} \|u_s\|_{\infty}) \rho_{V_{G,\phi,t_0}}(y) dy}{y^{2(n_1-a+1)}} $$

The RHS is in $L^1([M, +\infty[)$ by integration by parts as it was done in the proof of Lemma B.4. We conclude by doing the same on $]-\infty, -M]$.

**Proposition B.8** With the choice of potential $V_{\phi,t}$, the following map is continuous

$$ t \in [0, 1] \mapsto (c(\Xi_1^{-1}, H^n), c(\Xi_1^{-1}, W_n^\infty), c(\Theta^{(a)} \circ \Xi_1^{-1}, H^n)) $$

**Proof** By recalling the expression of those constants in (47), (57), (64), since in this appendix, all the building blocks in these constants were shown to be continuous we can conclude. □

## B.2 Parameter-continuity of $C_{\mathcal{L}_t}$ and $K_{V_{G,\phi,t}}$

In [DGM23][App. A], the authors showed that for a general potential $V$, the operator $\mathcal{A}$ considered as an unbounded operator on $H$ has the same spectrum as the Schrödinger operator $S: D(S) \to L^2(\mathbb{R})$, defined by

$$ S := -\Delta + w_V \quad \text{with} \quad D(S) := \left\{ u \in H^1(\mathbb{R}), uV' \in L^2(\mathbb{R}), -u'' + w_V u \in L^2(\mathbb{R}), \int_{\mathbb{R}} u(x) dx = 0 \right\} $$

and

$$ w_V := \frac{1}{2} \left( \frac{1}{2} V'^2 - V'' + 2PV'\mathcal{H}[\rho_V] - 2P\mathcal{H}[\rho_V'] + 2P^2\mathcal{H}[\rho_V]^2 \right) = \frac{1}{2} \left[ (\ln \rho_V)'' + \frac{1}{2} (\ln \rho_V)^2 \right]. \quad (98) $$

Since $C_{\mathcal{L}} = \lambda_1(\mathcal{A})^{-1/2} = \lambda_1(S)^{-1/2}$ by Theorem 4.3, we just have to show that when choosing the potential $V = V_{G,\phi,t}$, the $t$-dependent Schrödinger operator $S_t$ with potential $w_{V_{G,\phi,t}}$ produces a continuous smallest eigenvalue $\lambda_1(S_t)$. Before that, we recall the essential material for manipulating $S_t$.

**Proposition B.9** The map $t \mapsto C_{\mathcal{L}_t} = \lambda_1(S_t)^{-1/2}$ is continuous.

**Proof** First for all $t > 0$, $\lambda_1(S_t) > 0$. Secondly, we have the following equalities:

$$ E_t = \min_{\substack{u \in D(S_t) \\ \|u\|_2 = 1}} \langle u, S_t[u] \rangle_{L^2(\mathbb{R})} = \inf_{\substack{u \in C_c^\infty(\mathbb{R}) \\ \|u\|_2 = 1}} \int_{\mathbb{R}} (u')^2 + \int_{\mathbb{R}} u^2 w_{V_{G,\phi,t}}. $$

From the previous section $w_{V_{t'}}$ converges uniformly to $w_{V_{G,\phi,t}}$ when $t'$ goes to $t$. Hence for all $t, t' \in [0, 1]$, $u \in C_c^\infty(\mathbb{R})$ with $\|u\|_2 = 1$

$$ \left| \int_{\mathbb{R}} (u')^2 + \int_{\mathbb{R}} u^2 w_{V_{G,\phi,t}} - \int_{\mathbb{R}} (u')^2 + \int_{\mathbb{R}} u^2 w_{V_{t'}} \right| \le \|w_{V_{G,\phi,t}} - w_{V_{t'}}\|_{L^\infty(\mathbb{R})} $$

hence $\sup_{u \in C_c^\infty(\mathbb{R})} \left| \int_{\mathbb{R}} (u')^2 + \int_{\mathbb{R}} u^2 w_P^t - \int_{\mathbb{R}} (u')^2 + \int_{\mathbb{R}} u^2 w_{V_{t'}} \right| \to 0$ as $t'$ goes to $t$. Since uniform convergence is enough to ensure convergence of infinimums we get the result. □