# On the equilibrium measure for the Lukyanov integral

Charlie Dworaczek Guera, Karol K Kozlowski

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Charlie Dworaczek Guera, Karol K Kozlowski. On the equilibrium measure for the Lukyanov integral. 2024. [https://arxiv.org/abs/2406.04237](https://arxiv.org/abs/2406.04237)

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# On the equilibrium measure for the Lukyanov integral

Charlie Dworaczek Guera<fnref n="1" />

Univ Lyon, ENS de Lyon, Univ Claude Bernard Lyon 1, CNRS, Unité de Mathématiques Pures et Appliquées, F-69342 Lyon, France

Karol K. Kozlowski<fnref n="2" />

Univ Lyon, ENS de Lyon, Univ Claude Bernard Lyon 1, CNRS, Laboratoire de Physique, F-69342 Lyon, France

## Abstract

In 2000, Lukyanov conjectured that a certain ratio of N-fold integrals should provide access, in the large-N regime, to the ground state expectation value of the exponential of the Sinh-Gordon quantum field in 1+1 dimensions and finite volume R. This work aims at rigorously constructing the fundamental objects necessary to address the large-N analysis of such integrals. More precisely, we construct and establish the main properties of the equilibrium measure minimising a certain N-dependent energy functional that naturally arises in the study of the leading large-N behaviour of the Lukyanov integral. Our construction allows us to heuristically advocate the leading term in the large-N asymptotic behaviour of the mentioned ratio of Lukyanov integrals, hence supporting Lukyanov's prediction -obtained by other means- on the exponent σ of the power-law N^σ term of its asymptotic expansion as N → +∞.

MSC Classification: 82B23, 81Q80, 47B35, 60F10

## Contents

<table><tr><td><b>1 Introduction and statement of results</b></td><td><b>2</b></td></tr><tr><td>1.1 The separation of variable integral for the exponent of the field</td><td>2</td></tr><tr><td>1.2 The main results</td><td>4</td></tr><tr><td><b>2 The N-dependent equilibrium measure</b></td><td><b>7</b></td></tr><tr><td>2.1 General properties of the equilibrium measure</td><td>7</td></tr><tr><td>2.2 A truncated Wiener-Hopf based representation for the equilibrium measure</td><td>9</td></tr><tr><td>2.3 A convenient representation for the inverse acting on V'<sub>N;α</sub></td><td>15</td></tr><tr><td>2.4 Support of the equilibrium measure</td><td>18</td></tr></table>

<footnote n="1">¹e-mail: charlie.dworaczek@ens-lyon.fr</footnote>

<footnote n="2">²e-mail: karol.kozlowski@ens-lyon.fr</footnote>
# 1 Introduction and statement of results

## 1.1 The separation of variable integral for the exponent of the field

The seminal works of Al. Zamolodchikov [23], following the pioneering considerations of Yang-Yang [22], introduced the concept of Thermodynamic Bethe Ansatz (TBA) as a key tool allowing one to describe the ground state energies of integrable quantum field theories in finite volume. The construction, as an input, utilises the model's S-matrix and describes the per-volume ground state energy in terms of a solution to a non-linear integral equation. In the case of the Sinh-Gordon 1+1 dimensional quantum field theory, the TBA description was conjectured simultaneously and independently by Al. Zamolodchikov [24] and Lukyanov [19]. In that case, there is a unique TBA equation which takes the form

$$ \varepsilon(\lambda) = 2r \sin\left[\frac{\pi}{1+b^2}\right] \cosh(\lambda) + \int_{\mathbb{R}} K(\lambda - \mu) \ln\left[1 + e^{-\varepsilon(\mu)}\right] \quad \text{with} \quad K(\lambda) = \frac{4 \cosh(\lambda) \sin\left[\frac{\pi}{1+b^2}\right]}{\cosh(2\lambda) - \cos\left[\frac{2\pi}{1+b^2}\right]} \quad (1.1) $$

This equation involves two parameters r, b > 0. r = CmR with C > 0 some constant, R the model's volume and m the mass parameter. Finally, b > 0 measures the interaction strength.

It was rigorously shown in [14] that, for any r > 0, the non-linear integral equation (1.1) admits a unique solution in $L^\infty(\mathbb{R})$. With the solution at hand, the per-volume ground state energy admits the integral representation [24]

$$ -m \int_{\mathbb{R}} \frac{d\lambda}{2\pi} \cosh(\lambda) \ln\left[1 + e^{-\varepsilon(\lambda)}\right]. \quad (1.2) $$

These TBA-like considerations were backed up by Bethe Ansatz calculations carried out for the lattice discretisation of the finite-volume Sinh-Gordon quantum field theory in [8, 21].

By invoking an analogy with the classical method of separation of variables, Lukyanov [19] conjectured that the ground state expectation value of the exponential of the quantum Sinh-Gordon field $e^{\alpha\varphi}$ may be deduced from the data contained in the large-N behaviour of the below integral

$$ \delta_N[V_\alpha] = \int_{\mathbb{R}^N} d^N\lambda \prod_{k<\ell}^{N} \left\{ \sinh\left[(1+b^2)(\lambda_k - \lambda_\ell)\right] \cdot \sinh\left[(1+b^{-2})(\lambda_k - \lambda_\ell)\right] \right\} \prod_{k=1}^{N} e^{-V_\alpha(\lambda_k)}. \quad (1.3) $$

The potential $V_\alpha$ appearing above was expressed in terms of the solution $\varepsilon$ to the TBA equation for the Sinh-Gordon model (1.1) as

$$ V_{\alpha}(\lambda) = r \cosh(\lambda) - \alpha \lambda - \int_{\mathbb{R}} \frac{d\mu}{2\pi} \cdot \frac{g(\mu)}{\cosh(\lambda - \mu)} \quad \text{with} \quad g(\mu) = 2 \ln \left[1 + e^{-\varepsilon(\mu)}\right]. \quad (1.4) $$

It was conjectured in [19] that, as $N \to +\infty$,

$$ \frac{\delta_N[V_{\alpha}(b+b^{-1})]}{\delta_N[V_0]} = \left(\frac{N}{r}\right)^{\frac{\alpha^2}{2}} \cdot \langle e^{\alpha\varphi} \rangle_r \cdot (1 + o(1)). \quad (1.5) $$
The constant term in these asymptotics $\langle e^{\alpha\varphi} \rangle_r$ was conjectured to coincide with the ground state expectation value of the exponent of the properly normalised Sinh-Gordon quantum field in the finite volume $R$ theory. In this description, the fields were normalised so that, as $r \to +\infty$, the two-point functions of the fields have trivial CFT-like normalisation in the short space-like Minkowski-distance regime, see [19] for more details.

In fact, Lukyanov's integral falls into a much larger class of $N$-fold integrals describing so-called form factors -i.e. matrix elements- of local operators in numerous quantum integrable models solvable by the quantum separation of variables method, see e.g. [2, 11, 12, 13, 15, 16, 17, 18, 20]. In such a setting the form factors are expressed as a ratio $\mathcal{Z}[W_N]/\mathcal{Z}[V_N]$ for certain potentials $V_N, W_N$ and where

$$ \mathcal{Z}[V_N] = \int_{\mathcal{C}^N} d^N \lambda \prod_{k<l}^{N} \left\{ \sinh[\pi\omega_1(\lambda_k - \lambda_l)] \cdot \sinh[\pi\omega_2(\lambda_k - \lambda_l)] \right\} \prod_{k=1}^{N} e^{-V_N(\lambda_k)}. \qquad (1.6) $$

Above $\omega_1, \omega_2$ are related to a given model's coupling constants, and $\mathcal{C}$ is a model dependent curve in $\mathbb{C}$. The curve $\mathcal{C}$ may or may not be compact or closed. Also, the potential $V_N$ may or may not depend on $N$. However, typically, it is not varying with $N$ as $V_N(\lambda) = NU(\lambda)$ for some $N$ independent function $U$, i.e. the $N$-dependence, if present, is much more involved. For such multiple integrals, one is usually interested in the $N \to +\infty$ regime which allows one either to reach the thermodynamic or the continuum limit of the model. Thus, on top of testing Lukyanov's conjecture, the possibility to study of the large-$N$ behaviour of this class of integrals will have numerous applications in the field of quantum integrable models. We should mention that the above integral falls into the class of bi-orthogonal ensembles [3].

It is clear that the class of quantum separation of variables integrals $\mathcal{Z}[V_N]$ bears a strong structural resemblance with the spectral partition function of a random Hermitian matrix $M$ sampled from a measure $e^{-N\text{Tr}[V(M)]}dM$, with $dM$ being a properly normalised Lebesgue measure on the statistically independent entries. Indeed, the latter takes the form

$$ \mathcal{Z}_{N; \text{Herm}}[V] = \int_{\mathbb{R}^N} d^N \lambda \prod_{k<l}^{N} |\lambda_k - \lambda_l|^2 \prod_{k=1}^{N} e^{-NV(\lambda_k)} \qquad (1.7) $$

In both cases (1.6)-(1.7), there appears a one-body confining potential and a repulsive two-body interaction vanishing as the square of the spacing between the integration variables. In fact, the two-body interaction is given by Vandermonde determinants in both cases: the square of a usual Vandermonde in the random matrix case (1.7) and the product of two-hyperbolic Vandermondes in the quantum separation of variables case (1.6). One could thus hope that the techniques allowing one to deal with the large-$N$ behaviour of classical random matrix ensembles will also be fit for tackling the large-$N$ behaviour of quantum separation of variables issued integrals. Unfortunately, the situation is way more intricate and takes its origin in crucial differences between these two types of integrals. Genuinely, the potentials arising in the quantum separation of variables case are not varying with $N$ as $V_N(\lambda) = NU(\lambda)$ for some $N$ independent function $U$. Thus, the two-body and the one-body interactions in $\mathcal{Z}[V_N]$ evolve on different scales and one needs to dilatate the integration variables, in an appropriate fashion, so that both rescaled interactions equilibrate. While in the random matrix case the two-body interaction was behaving trivially under rescalings, this is not anymore the case in the quantum separation of variables setting. This introduces several additional scales in $N$ to the problem what makes numerous of the steps developed for the random matrix case very tricky, technically speaking, to set in. We would like to mention that, in fact, certain instances of integrals of the type (1.6) did in fact appear directly in the random matrix literature. More precisely, the spectral

<footnote>†Else, indeed, the study of the large-$N$ behaviour of quantum separation of variables issued multiple integrals would follow from the application of techniques developed [6]</footnote>
part of a random Hermitian matrix's sampled from a measure $\propto e^{-N\text{Tr}[V(M)-AM]}dM$ with $A = \text{diag}(a_1, \dots, a_N)$, $a_k = (k-1)/N$, admits the integral representation [9]

$$ \mathcal{Z}_{N;\text{Source}}[V] = \int_{\mathbb{R}^N} d^N \lambda \prod_{k<\ell}^{N} \{ (\lambda_k - \lambda_\ell) \sinh(\lambda_k - \lambda_\ell) \} \prod_{k=1}^{N} e^{-NV(\lambda_k)}. \qquad (1.8) $$

The asymptotic expansion of such integrals may be dealt with by using the techniques developed in [6]. The work [9] proposed a Riemann-Hilbert approach for bi-orthogonal polynomials that could, in principle, allow one to extract the large-$N$ behaviour of (1.8) and, more generally, (1.6). However, in the case of the quantum separation of variables issued integral, the necessity for rescaling the integration variables would introduce numerous technical complications to the large-$N$ analysis technique developed in [9]. In particular, it would demand to have a highly detailed control on the $N$-dependent equilibrium measure that will be obtained in the present paper.

The first progress in this direction of achieving a large-$N$ analysis of multiple integrals of the form (1.6) was achieved in [7] where techniques allowing one to deal with $N$-dependent two-body interactions were developed. The aim of this work is to push further the results obtained in [7] and lay the ground for rigorously justifying the presence of the power-law term in $N$ in (1.5), and, in a second stage, for obtaining rigorously the whole expansion up to o(1). In order to apply concentration of measure techniques which were first developed for $\beta$-ensembles in [1] and, later, extended so as to allow to deal with more complex integrals in [4, 6, 7], one first needs to have a good grasp on the so-called equilibrium measure. As explained in [7], in the case of the partition function (1.3), the latter corresponds to the unique minimiser of an $N$-dependent functional on $M^1(\mathbb{R})$, the space of probability measures on $\mathbb{R}$. The construction of the equilibrium measure is the main achievement of this work. As mentioned, this may also pave the way to the Riemann-Hilbert analysis of the large-$N$ behaviour of (1.3) by means of bi-orthogonal polynomials. Our result allows us to back up the prediction on the leading large-$N$ behaviour given in (1.5), although the lack of sharp estimates on the remainders does not allow us to turn our findings into a rigorous proof.

The paper is organised as follows. In Subsection 1.2 we gather the main results obtained in this work. Section 2 is devoted to the construction of the equilibrium measure governing the leading large-$N$ behaviour of the Lukyanov integral. In Subsection 2.1, we establish basic properties of the equilibrium measure. In Subsection 2.2, we recall the explicit representation for the inverse of a truncated Wiener-Hopf operator that arises in the characterisation of the equilibrium measure's density. In Subsection 2.3 we establish a convenient integral representation for the equilibrium measure's density. Finally, in Subsection 2.4, we construct its support. Section 3 is devoted to obtaining the first few terms in the large-$N$ asymptotic expansion of a certain integral versus the equilibrium measure that plays a role in the problem of our interest.

## 1.2 The main results

It is easy to see that the repulsive nature of the sinh two-body interaction and the confining nature of the potential are of the same order of magnitude in $N$ on a scale $\ln N$. Hence, so as to deal with finite quantities, it appears convenient to rescale the integration variables in (1.3) by $\ln N$. Then, it holds $\beta_N[V_\alpha] = [\ln N]^N \mathcal{Z}_N[V_{N;\alpha}]$ with

$$ \mathcal{Z}_N[V_{N;\alpha}] = \int_{\mathbb{R}^N} d^N \lambda \prod_{a<b}^{N} \left\{ \sinh\left[\frac{\bar{\omega}_1}{2}(\lambda_a - \lambda_b)\right] \cdot \sinh\left[\frac{\bar{\omega}_2}{2}(\lambda_a - \lambda_b)\right] \right\} \prod_{a=1}^{N} e^{-N\tau_N V_{N;\alpha}(\lambda_a)}. \qquad (1.9) $$

The two periods $\bar{\omega}_a$ grow with $N$ as

$$ \bar{\omega}_a = 2\pi\tau_N\omega_a \quad \text{with} \quad \tau_N = \ln N. \qquad (1.10) $$
The rescaled confining potential takes the form

$$V_{N;\alpha}(\lambda) = \frac{r}{N\tau_N} \cosh[\tau_N\lambda] - \frac{\alpha\lambda}{N} - \int_{\mathbb{R}} \frac{d\mu}{2\pi N} \cdot \frac{g(\tau_N\mu)}{\cosh[\tau_N(\lambda - \mu)]}. \quad (1.11)$$

The connection to the Lukyanov integral imposes the following from for the periods

$$\omega_1 = \frac{1 + b^2}{\pi} \quad \text{and} \quad \omega_2 = \frac{1 + b^{-2}}{\pi}. \qquad (1.12)$$

Following the techniques of [1] and their adaptation to the N-dependent setting developed in [7], one may show that

$$\mathcal{Z}_N[V_{N;\alpha}] = \exp\left\{-N^2 \tau_N \inf_{\mu \in \mathcal{M}^1(\mathbb{R})} \{\mathcal{E}_N[\mu]\} + O(N\tau_N^2)\right\}, \quad (1.13)$$

in which the N-dependent functional on $\mathcal{M}^1(\mathbb{R})$ takes the form

$$\mathcal{E}_N[\mu] = \int_{\mathbb{R}} d\mu(s)V_{N;\alpha}(s) - \frac{1}{2\tau_N} \int_{\mathbb{R}} d\mu(s)d\mu(t) \ln \left\{ \prod_{a=1}^{2} \sinh \left( \bar{\omega}_a \frac{s-t}{2} \right) \right\}. \quad (1.14)$$

$\mathcal{E}_N$ is strictly convex, lower-continuous and has compact level sets, see [7] for more details. As such, it admits a unique minimiser on $\mathcal{M}^1(\mathbb{R})$ denoted by $\tilde{\mu}_{\text{eq};\alpha}$. The knowledge of this minimum thus allows one to access the first few terms in the asymptotic expansion of $\ln \mathcal{Z}_N[V_{N;\alpha}]$.

Our main result is gathered in the

**Theorem 1.1.** The equilibrium measure $\tilde{\mu}_{\text{eq};\alpha}$ is Lebesgue continuous with a density $\tilde{\rho}_{\text{eq};\alpha}$ given by the square root of an analytic function. There exists $r_0$ such that, for any $N$ and $r \ge r_0$, it is supported on the segment $\sigma_{N;\alpha} = [a_{N;\alpha}; b_{N;\alpha}]$. Moreover, there exists $N_0$ such that, for any $N \ge N_0$ it is given by

$$\tilde{\rho}_{\text{eq};\alpha} = \mathcal{W}_N[V'_{N;\alpha}\mathbf{1}_{\sigma_{N;\alpha}}]_{\substack{a_N \leftarrow a_{N;\alpha} \\ b_N \leftarrow b_{N;\alpha}}} \quad (1.15)$$

where $\mathcal{W}_N$ is the integral transform given by (2.23) in which one should specify the periods $\omega_a$ as in (1.12).

The endpoints $a_{N;\alpha}$, $b_{N;\alpha}$ of the support admit the large-N expansion

$$\tau_N b_{N;\alpha} = \ln\left(\frac{\mathfrak{d}_0 N}{2}\right) + \frac{\alpha}{N(1+b^2)(1+b^{-2})} \\ + \frac{1}{N^2} \left\{ \mathfrak{d}_1 \left(1 + \frac{2\mathcal{F}[g](i)}{\pi r} \mathbf{1}_{1<\zeta}\left(1+\frac{\alpha}{\pi}\right)\right) - \frac{\alpha^2}{2(1+b^2)^2(1+b^{-2})^2} \right\} + \mathcal{O}\left(\frac{1}{N^{3-\eta}}\right) \quad (1.16)$$

and

$$\tau_N a_{N;\alpha} = -\ln\left(\frac{\mathfrak{d}_0 N}{2}\right) + \frac{\alpha}{N(1+b^2)(1+b^{-2})} \\ - \frac{1}{N^2} \left\{ \mathfrak{d}_1 \left(1 + \frac{2\mathcal{F}[g](i)}{\pi r} \mathbf{1}_{1<\zeta}\left(1+\frac{\alpha}{\pi}\right)\right) + \frac{\alpha^2}{2(1+b^2)^2(1+b^{-2})^2} \right\} + \mathcal{O}\left(\frac{1}{N^{3-\eta}}\right) \quad (1.17)$$

where $\eta > 0$ is fixed but can be taken as small as need be. The two constants $\mathfrak{d}_0$, $\mathfrak{d}_1$ arising in this expansion take the form

$$\mathfrak{d}_0 = \frac{2}{r\sqrt{\pi}} \prod_{\nu=\pm} \left\{ (1 + b^{2\nu})^{\frac{-1}{2(1+b^{2\nu})}} \cdot \Gamma\left(\frac{1}{2(1+b^{2\nu})}\right) \right\} \quad \text{and} \quad \mathfrak{d}_1 = \frac{r^2}{\pi} \prod_{\nu=\pm} \left\{ \frac{\sin\left[\frac{\pi}{2(1+b^{2\nu})}\right]}{1+b^{2\nu}} \right\}. \quad (1.18)$$
The hardest part of the theorem consists in proving the form of the endpoints, the rest of its content follows from the techniques already developed in [7].

One way to obtain the large-$N$ behaviour of the ratio: $\beta_N[V_\alpha]/\beta_N[V_0]$, i.e. $\mathcal{Z}_N[V_{N;\alpha}]/\mathcal{Z}_N[V_{N;0}]$ is to start from the differential identity

$$ \partial_{\alpha} \ln \mathcal{Z}_N[V_{N;\alpha}] = N \tau_N \mathbb{E}_{N;\alpha} \left[ \int \xi dL_N^{(\lambda_N)}(\xi) \right] \quad \text{with} \quad L_N^{(\lambda_N)} = \frac{1}{N} \sum_{a=1}^{N} \delta_{\lambda_a} \qquad (1.19) $$

being the empirical distribution of the integration variables, $\lambda_N = (\lambda_1, \dots, \lambda_N)$, and $\mathbb{E}_{N;\alpha}$ referring to the expectation in respect to the probability measure $\mathbb{P}_{N;\alpha}$ on $\mathbb{R}^N$ with density

$$ \mathbb{P}_{N;\alpha}(\lambda_N) = \frac{1}{\mathcal{Z}_N[V_{N;\alpha}]} \prod_{a<b}^{N} \left\{ \sinh\left[\frac{\bar{\omega}_1}{2}(\lambda_a - \lambda_b)\right] \cdot \sinh\left[\frac{\bar{\omega}_2}{2}(\lambda_a - \lambda_b)\right] \right\} \prod_{a=1}^{N} e^{-N\tau_N V_{N;\alpha}(\lambda_a)}. \qquad (1.20) $$

Using concentration of measure techniques as in [7] one may show that under $\mathbb{P}_{N;\alpha}$ the empirical measure concentrates around the equilibrium measure in the sense that for smooth functions growing at most polynomially at infinity it holds

$$ \left| \mathbb{E}_{N;\alpha} \left[ \int \phi(\xi) d(\hat{\mu}_{\text{eq};\alpha} - L_N^{(\lambda_N)})(\xi) \right] \right| \le C \frac{\tau_N}{\sqrt{N}}. \qquad (1.21) $$

for some constant depending on $\phi$. This reasoning then entails that

$$ \partial_{\alpha} \ln \mathcal{Z}_N[V_{N;\alpha}] = N \tau_N \int \xi d\hat{\mu}_{\text{eq};\alpha}(\xi) + O(\sqrt{N} \tau_N^2). \qquad (1.22) $$

The remainder is uniform in $\alpha$. It is important to stress that the bounds issuing from the concentration of measure estimates are only a priori bounds. These may be improved by using the machinery of loop equations which allow one to improve the rigidity of the fluctuations, see [5] for an implementation of the method in the case of $\beta$-ensembles, and compute the subdominant corrections contained in the O remainder above. However, in the case of the present multiple integral, there arise several technical difficulties in the analysis of the associated system of loop equations which go beyond the scope of the present analysis. We would however like to point out that under the specialisation (1.12), it holds

$$ N \tau_N \int \xi d\hat{\mu}_{\text{eq};\alpha}(\xi) = \frac{\alpha \ln N}{(1 + b^2)(1 + b^{-2})} + O(1), \qquad (1.23) $$

with a remainder that is uniform in $\alpha$. See Proposition 3.1 for the detailed statement.

This leads to the suggestive results

$$ \ln \left( \frac{\mathcal{Z}_N[V_{N;\alpha(b+b^{-1})^2}]}{\mathcal{Z}_N[V_{N;0}]} \right) = \frac{\alpha^2}{2} \ln N + \mathfrak{R}_N \qquad (1.24) $$

with $\mathfrak{R}_N$ a remainder that we are able only to estimate as $O(\sqrt{N}\tau_N^2)$. However, we expect that this is an overestimate and that eventually, the machinery of loop equations will allow us to prove that $\mathfrak{R}_N = O(1)$. Thus, while not being a rigorous proof thereof, the above provides a strong check of Lukyanov's conjecture.
## 2 The N-dependent equilibrium measure

From now on, we keep $\omega_1, \omega_2$ arbitrary and shall specify our results to the setting (1.12) only at the very end.

### 2.1 General properties of the equilibrium measure

Consider the multiple integral

$$ \mathcal{X}_M = \int_{\mathbb{R}^M} d^M \lambda \prod_{a<b}^{M} \left\{ \sinh[\pi\omega_1\tau_N(\lambda_a - \lambda_b)] \cdot \sinh[\pi\omega_2\tau_N(\lambda_a - \lambda_b)] \right\}^{\frac{1}{\tau_N}} \prod_{a=1}^{N} e^{-MV_{N;\alpha}(\lambda_a)}. \quad (2.1) $$

There $N$ is fixed and to be considered as an outer parameter. This kind of integral has been studied in [7]. It was shown there that

$$ \lim_{M \to +\infty} \left\{ \frac{1}{M^2} \ln \mathcal{X}_M \right\} = - \inf_{\mu \in \mathcal{M}(\mathbb{R})} \mathcal{E}_N[\mu] \quad (2.2) $$

with $\mathcal{E}_N$ as defined in (1.14). The minimum is attained at a unique measure $\hat{\mu}_{eq;\alpha}$ that has compact support given by a finite union of segments and is Lebesgue continuous with a density $\widehat{\rho}_{eq;\alpha}$ given by the square root of an function analytic in an open neighbourhood of the support$^†$. In particular, the density is smooth in the interior of the support and vanishes at least as a square root at the edges of the support.

**Lemma 2.1.** There exists $r_0 > 0$ such that, for any $r \ge r_0$, the equilibrium measure has connected support

$$ \sigma_{N;\alpha} = \supp[\hat{\mu}_{eq;\alpha}] = [a_{N;\alpha}; b_{N;\alpha}]. \quad (2.3) $$

**Proof—**

It is a classical fact that the support of the equilibrium measure will be connected, viz. of the form (2.3), as soon as $V_{N;\alpha}$ is strictly convex, see e.g. Appendix C of [7]

A direct calculation starting from (1.11) yields

$$ V''_{N;\alpha}(\lambda) = \frac{r\tau_N}{N} \cosh[\tau_N\lambda] - \tau_N^2 \int_{\mathbb{R}} \frac{d\mu}{2\pi N} \cdot \left\{ \frac{1}{\cosh[\tau_N(\lambda - \mu)]} - 2 \frac{\sinh^2[\tau_N(\lambda - \mu)]}{\cosh^3[\tau_N(\lambda - \mu)]} \right\} \cdot g(\tau_N\mu). \quad (2.4) $$

It is easy to infer from the non-linear integral equation satisfied by $\varepsilon$, that there exist $r$-independent $c_\varepsilon, c'_\varepsilon > 0$ such that

$$ e^{-\varepsilon(\lambda)} \le c_{\varepsilon} e^{-\tau c'_{\varepsilon} \cosh(\lambda)}. \quad (2.5) $$

One thus gets the lower bound

$$ \begin{aligned} V''_{N;\alpha}(\lambda) \ge \frac{r\tau_N}{N} \cosh[\tau_N\lambda] - c_\varepsilon \tau_N^2 e^{-\tau c'_\varepsilon} \int_{\mathbb{R}} \frac{d\mu}{2\pi N} \cdot & \left| \frac{1}{\cosh[\tau_N\mu]} - 2 \frac{\sinh^2[\tau_N\mu]}{\cosh^3[\tau_N\mu]} \right| \\ & \ge \frac{\tau_N}{N} \{r - C e^{-\tau c'_\varepsilon}\} > 0 \quad (2.6) \end{aligned} $$

where the last bound follows provided that $r \ge r_0$ for some $r_0 > 0$.

<footnote>$^†$In our $N$ dependent setting, the size of this neighbourhood will naturally depend on $N$</footnote>
It is a standard fact, see e.g. [10] that $\hat{\mu}_{eq;\alpha}$ corresponds to the unique solution to the variational problem

$$V_{N;\alpha}(\lambda) - \frac{1}{\tau_N} \int d\hat{\mu}_{eq;\alpha}(s) \ln \left[ \prod_{a=1}^{2} \sinh \left( \bar{\omega}_a \frac{\lambda - s}{2} \right) \right] \begin{cases} = C_{eq;\alpha}^{(N)} & \lambda \in \sigma_{N;\alpha} \\ > C_{eq;\alpha}^{(N)} & \lambda \in \mathbb{R} \setminus \sigma_{N;\alpha} \end{cases}. \quad (2.7)$$

First of all, due to the smoothness of the equilibrium's measure density and its square root vanishing at the edges, we could formulate this problem in the strong sense, i.e. pointwise, and not a.e.. Also, we stress that due to the strict convexity of $V_{N;\alpha}$ for $r \ge r_0$, the second condition is immediately satisfied, see Appendix C of [7] for more details.

This variational characterisation of the equilibrium measure allows one to obtain a priori upper/lower bounds on the endpoints $a_{N;\alpha}/b_{N;\alpha}$.

**Lemma 2.2.** There exists $N_0 > 0$ and $\zeta > 0$ such that, for any $N \ge N_0$,

$$a_{N;\alpha} \le -\zeta \quad \text{and} \quad b_{N;\alpha} \ge \zeta. \qquad (2.8)$$

*Proof—*

The proof goes by contradiction. Thus assume that for any $\zeta > 0$ and $N_0$ there exists $N \ge N_0$ such that

$$a_{N;\alpha} \ge -\zeta \quad \text{or} \quad b_{N;\alpha} \le \zeta. \qquad (2.9)$$

One may take $\zeta < 1/4$ and extracting sub-sequences if need be, one gets that there exists a sequence $N_k \to +\infty$ such that, for any $k$, $b_{N_k;\alpha} \le \zeta$ or, for any $k$, $a_{N_k;\alpha} \ge -\zeta$. We discuss in detail the first case scenario, the second one can be excluded in the same fashion.

We start by introducing the so-called effective potential

$$V_{\text{eff}}(\lambda) = V_{N;\alpha}(\lambda) + \int d\hat{\mu}_{\text{eq};\alpha}(s) \hat{f}_N(\lambda, s) \quad \text{with} \quad \hat{f}_N(\lambda, s) = \frac{-1}{\tau_N} \ln \left[ \prod_{a=1}^{2} \sinh \left( \bar{\omega}_a \frac{\lambda - s}{2} \right) \right]. \quad (2.10)$$

One has that

$$\partial_{\lambda} \hat{f}_N(\lambda, s) = - \sum_{a=1}^{2} \pi \omega_a \coth[\pi \omega_a \tau_N(\lambda - s)] < 0 \quad \text{for} \quad \lambda \ge s, \qquad (2.11)$$

which entails that $\hat{f}_N(\lambda, s) - \hat{f}_N(\mu, s) < 0$ for any $\lambda > \mu \ge s$. One may thus decompose, for any $1/2 > \lambda > 1/4$,

$$V_{\text{eff}}(\lambda) - V_{\text{eff}}(b_{N_k;\alpha}) = V_{N_k;\alpha}(\lambda) - V_{N_k;\alpha}(b_{N_k;\alpha}) + \int d\hat{\mu}_{\text{eq};\alpha}(s) (\hat{f}_{N_k}(\lambda, s) - \hat{f}_{N_k}(\frac{1}{4}, s)) \\ + \int d\hat{\mu}_{\text{eq};\alpha}(s) (\hat{f}_{N_k}(\frac{1}{4}, s) - \hat{f}_{N_k}(b_{N_k;\alpha}, s)). \quad (2.12)$$

One may then bound each term as follows. Since $1/4 > \zeta \ge b_{N_k;\alpha}$, the last line produces a purely negative contribution. Further, it is direct to estimate that in the range of $\lambda$s considered $V_{N_k;\alpha}(\lambda) = o(1)$ as $k \to +\infty$. Likewise, if $-3/4 \le b_{N_k;\alpha} \le \zeta$ then one has that $V_{N_k;\alpha}(b_{N_k;\alpha}) = o(1)$, while for $b_{N_k;\alpha} \le -3/4$ it holds for $k$ large enough that $V_{N_k;\alpha}(b_{N_k;\alpha}) \ge V_{N_k;\alpha}(\lambda)$. Thus, whatever the regime, it holds

$$V_{N_k;\alpha}(b_{N_k;\alpha}) - V_{N_k;\alpha}(\lambda) \le o(1). \qquad (2.13)$$
Finally, one has, for $\lambda > s$ that

$$f_N(\lambda, s) = -\pi(\omega_1 + \omega_2)(\lambda - s) + O\left(\frac{1}{\tau_N}\sum_{a=1}^2 e^{-\bar{\omega}_a(\lambda-s)}\right). \qquad (2.14)$$

Thus, since $\lambda$ and $1/4$ are uniformly away from $s$ in the integral arising in the first line of (2.12), one gets that

$$V_{\text{eff}}(\lambda) - C_{\text{eq};\alpha}^{(N)} = V_{\text{eff}}(\lambda) - V_{\text{eff}}(b_{Nk};\alpha) \le o(1) - \pi(\omega_1 + \omega_2)(\lambda - \frac{1}{4}) < 0. \qquad (2.15)$$

Since $\lambda \notin \sigma_{N;\alpha}$, this contradicts the variational equation (2.7).

One knows from Lemma 2.1 that the equilibrium measure is supported on the segment $[a_{N;\alpha}; b_{N;\alpha}]$. Since $\widehat{\varrho}_{\text{eq};\alpha}$ is smooth on $]a_{N;\alpha}; b_{N;\alpha}[$ and admits at worst square root singularities at the edges $a_{N;\alpha}, b_{N;\alpha}$, one has that $\widehat{\varrho}_{\text{eq};\alpha} \in H_s([a_{N;\alpha}; b_{N;\alpha}])$ for any $0 < s < 1/2$. In particular, one may differentiate the first relation given in (2.7) what yields the singular-integral equation satisfied by the equilibrium measure's density

Now, Lemma 2.2 ensures that $b_{N;\alpha} - a_{N;\alpha} \ge 2\zeta > 0$, so that $\widehat{\varrho}_{\text{eq};\alpha}$ solves a truncated Wiener-Hopf equation in which the renormalised difference of boundaries satisfies

This allows one to invoke Riemann-Hilbert techniques so as to solve the equation in the large-$N$ regime and will provide the starting point for characterising the measure and its support thoroughly in the large-$N$ regime.

## 2.2 A truncated Wiener-Hopf based representation for the equilibrium measure

In the following, we denote by $H_s(\mathbb{R})$ the $s^{th}$ Sobolev space, viz.

Here, the Fourier transform $\mathcal{F}$ is defined, for $g \in L^1(\mathbb{R})$, as

Then, for any closed $F \subset \mathbb{R}$, $H_s(F)$ is the space of functions $f \in H_s(\mathbb{R})$ such that $\text{supp}[f] \subset F$.

It was established in [7] that, provided that $b_N - a_N > \eta$ for some $\eta > 0$ and that $N$ is large enough, the singular integral operator $S_N : H_s([a_N; b_N]) \to H_s(\mathbb{R})$ with $0 < s < 1/2$, defined for sufficiently regular functions $h$ by the integral transform

$$S_N[h](\lambda) = \sum_{a=1}^{2} \pi \omega_a \int d\mu h(\mu) \coth \left[ \bar{\omega}_a \frac{\lambda - \mu}{2} \right] \qquad (2.20)$$
is invertible on the co-dimension 1 subspace of $H_s(\mathbb{R})$

$$
\mathcal{X}_s(\mathbb{R}) = \{g \in H_s(\mathbb{R}) : \mathcal{J}[g] = 0\} \quad \text{with} \quad \mathcal{J}[g] = \int_{\mathbb{R}+i\epsilon'} \frac{d\mu}{2i\pi} \chi_{11}(\mu) e^{-i\bar{b}_{N\mu}\mathcal{F}[g]}(\tau_N\mu). \quad (2.21)
$$

Above and in the following, $\epsilon' > 0$ is arbitrary but taken sufficiently small and we agree upon

$$
x_N = b_N - a_N, \quad \bar{x}_N = \tau_N x_N, \quad \bar{b}_N = \tau_N b_N \quad \text{and} \quad \bar{a}_N = \tau_N a_N. \tag{2.22}
$$

Moreover, without further notice we shall assume in this subsection that the lower bound $b_N - a_N \ge \eta$ with $\eta > 0$
holds and that $N$ is large enough.

The closed subspace $\mathcal{X}_s(\mathbb{R})$ along with the inverse $\mathcal{W}_N$ are both described in terms of a piecewise holomorphic
$2 \times 2$ matrix valued function $\chi$ that we shall discuss below. First, however, we provide the expression for the inverse
$\mathcal{W}_N$. For any sufficiently regular $g \in \mathcal{X}_s(\mathbb{R})$, the latter takes the form of the integral transform

$$
\mathcal{W}_N[g](\xi) = \frac{\tau_N^2}{2\pi} \int_{\mathbb{R}+2i\epsilon'} \frac{d\lambda}{2i\pi} \int_{\mathbb{R}+i\epsilon'} \frac{d\mu}{2i\pi} \frac{e^{-i\tau_N\lambda(\xi-a_N)}}{\mu-\lambda} \{\chi_{11}(\lambda)\chi_{12}(\mu) - \frac{\mu}{\lambda}\chi_{11}(\mu)\chi_{12}(\lambda)\} e^{-i\bar{b}_{N\mu}\mathcal{F}[g](\tau_N\mu)}. \quad (2.23)
$$

in which $\epsilon' > 0$ can be taken as small as need be.

$\mathcal{W}_N$ and $\mathcal{J}$ are both expressed in terms of the unique solution $\chi$ to the $2 \times 2$ Riemann–Hilbert problem

• $\chi_{1a} \in O(\mathbb{C} \setminus \mathbb{R})$ and $\lambda \mapsto \lambda\chi_{2a}(\lambda) \in O(\mathbb{C} \setminus \mathbb{R})$ admit continuous $\pm$ boundary values on $\mathbb{R}$;

• $\chi_+(\lambda) = G_\chi(\lambda)\chi_-(\lambda)$, with a jump matrix

$$
G_{\chi}(\lambda) = \begin{pmatrix} e^{i\lambda\bar{x}_N} & 0 \\ R(\lambda) & -e^{-i\lambda\bar{x}_N} \end{pmatrix} \quad \text{where} \quad R(\lambda) = \frac{\sinh\left[\frac{\lambda}{2}\left(\frac{1}{\omega_1} + \frac{1}{\omega_2}\right)\right]}{2\sinh\left[\frac{\lambda}{2\omega_1}\right]\sinh\left[\frac{\lambda}{2\omega_2}\right]} \quad (2.24)
$$

• as $\lambda \to \infty$

$$
\chi(\lambda) = \begin{cases}
\begin{pmatrix} -\operatorname{sgn}[\mathcal{R}(\lambda)]e^{i\lambda\bar{x}_N} & 1 \\ -1 & 0 \end{pmatrix} \cdot [-i\lambda]^{-\frac{\sigma_3}{2}} \cdot (I_2 + \frac{\chi_1}{\lambda} + O(\frac{1}{\lambda^2})) , & \lambda \in \mathbb{H}^+ \\
\begin{pmatrix} -1 & \operatorname{sgn}[\mathcal{R}(\lambda)]e^{-i\lambda\bar{x}_N} \\ 0 & 1 \end{pmatrix} \cdot [i\lambda]^{-\frac{\sigma_3}{2}} e^{i\frac{\pi}{2}\sigma_3} \cdot (I_2 + \frac{\chi_1}{\lambda} + O(\frac{1}{\lambda^2})) , & \lambda \in \mathbb{H}^-
\end{cases} . \quad (2.25)
$$

It was established in [7] that $\chi$ satisfies the variable reflection relation

$$
\chi(-\lambda) = \begin{pmatrix} \chi_{11}(\lambda) & -\lambda\chi_{11}(\lambda) + \chi_{12}(\lambda) \\ -\chi_{21}(\lambda) & \lambda\chi_{21}(\lambda) - \chi_{22}(\lambda) \end{pmatrix}, \quad (2.26)
$$

the complex conjugation property

$$
(\chi(\lambda^*))^* = \begin{pmatrix} -\chi_{11}(-\lambda) & \chi_{12}(-\lambda) \\ \chi_{21}(-\lambda) & -\chi_{22}(-\lambda) \end{pmatrix}. \qquad (2.27)
$$

Again, it follows from [7] that $\chi$ admits an explicit large-$N$ asymptotic behaviour valid as soon as $b_N - a_N$ is
bounded away from zero uniformly in $N$. Below, we list the uniform large-$N$ asymptotic expansions in the regions
of $\mathbb{C}$ which are pertinent for our needs. These regions are delimited by the real axis and the curves $\Gamma_{\uparrow/\downarrow}$ as depicted
in Fig. 1. Note that the point i, resp. -i, may be below or above of $\Gamma_↑$, resp. $\Gamma_↓$, depending on the chosen range for $\omega_1, \omega_2$. First, however, we point out that $R$ admits a Wiener-Hopf-like factorisation

$$R(\lambda) = R_{\uparrow}(\lambda) R_{\downarrow}(\lambda) \qquad (2.28)$$

where

$$R_{\uparrow}(\lambda) = \frac{i}{\lambda} \cdot \sqrt{\omega_1 + \omega_2} \cdot \left(\frac{\omega_2}{\omega_1 + \omega_2}\right)^{\frac{i\lambda}{2\pi\omega_1}} \cdot \left(\frac{\omega_1}{\omega_1 + \omega_2}\right)^{\frac{i\lambda}{2\pi\omega_2}} \cdot \frac{\prod_{p=1}^{2} \Gamma\left(1 - \frac{i\lambda}{2\pi\omega_p}\right)}{\Gamma\left(1 - \frac{i\lambda(\omega_1 + \omega_2)}{2\pi\omega_1\omega_2}\right)} \qquad (2.29)$$

and

$$R_{\downarrow}(\lambda) = \frac{\lambda}{2\pi \sqrt{\omega_1 + \omega_2}} \cdot \left(\frac{\omega_2}{\omega_1 + \omega_2}\right)^{-\frac{i\lambda}{2\pi\omega_1}} \cdot \left(\frac{\omega_1}{\omega_1 + \omega_2}\right)^{-\frac{i\lambda}{2\pi\omega_2}} \cdot \frac{\prod_{p=1}^{2} \Gamma\left(\frac{i\lambda}{2\pi\omega_p}\right)}{\Gamma\left(\frac{i\lambda(\omega_1 + \omega_2)}{2\pi\omega_1\omega_2}\right)}. \qquad (2.30)$$

Note that

$$R_{\downarrow}(0) = -i \sqrt{\omega_1 + \omega_2} \quad \text{and} \quad (\lambda R_{\uparrow}(\lambda))_{|\lambda=0} = i \sqrt{\omega_1 + \omega_2}. \qquad (2.31)$$

Also, $R_↑$ and $R_↓$ satisfy to the relations

$$R_{\uparrow}(-\lambda) = \lambda^{-1} \cdot R_{\downarrow}(\lambda) \quad \text{and} \quad (R_{\uparrow}(\lambda^*))^* = \lambda^{-1} \cdot R_{\downarrow}(\lambda). \qquad (2.32)$$

Furthermore, $R_{\uparrow/\downarrow}$ admit the asymptotic behaviour

$$R_{\uparrow}(\lambda) = (-i\lambda)^{-\frac{1}{2}} \cdot (1 + O(\lambda^{-1})) \quad \text{for} \quad \lambda \to \infty \qquad (2.33)$$

$$R_{\downarrow}(\lambda) = -i(i\lambda)^{\frac{1}{2}} \cdot (1 + O(\lambda^{-1})) \quad \text{for} \quad \lambda \to \infty. \qquad (2.34)$$

The notation $\uparrow$ and $\downarrow$ indicates the direction, in respect to $\mathbb{R} + i\epsilon$ in the complex plane where $R_{\uparrow/\downarrow}$ have no pole nor zeroes.

The mentioned uniform asymptotic expansions involve an auxiliary, piecewise analytic, matrix

$$\Pi(\lambda) = \mathbb{I}_2 + O\left(\frac{e^{-\zeta(1-\eta)\bar{\chi}_N}}{1 + |\lambda|}\right) \quad \text{with} \quad \zeta = \frac{2\pi\omega_1\omega_2}{\omega_1 + \omega_2}. \qquad (2.35)$$

This bound holds for any fixed $\eta > 0$ and uniformly on $\mathbb{C}$. Moreover, the remainder is smooth in $a_N$ and $b_N$ with derivatives controlled as

$$\partial_{a_N}^k \partial_{b_N}^l \Pi(\lambda) = \mathbb{I}_2 \delta_{k;0} \delta_{l;0} + O\left(\tau_N^{k+l} \frac{e^{-\zeta(1-\eta)\bar{\chi}_N}}{1 + |\lambda|}\right) \qquad (2.36)$$

Finally, one has

$$P_R(\lambda) = \mathbb{I}_2 + \frac{\tilde{\vartheta}_R}{\lambda} \Pi^{-1}(0) \sigma^{-} \Pi(0) \quad \text{with} \quad \tilde{\vartheta}_R = \frac{1}{1 - [\Pi'(0)\Pi^{-1}(0)]_{12}}. \qquad (2.37)$$
The entries $\chi_{11}, \chi_{12}$ admit holomorphic continuations from $\mathbb{H}^{\pm}$ into some small tubular neighbourhood of $\mathbb{R}$ in $\mathbb{H}^{\pm}$. In particular, $\chi_{11;\pm}(\lambda), \chi_{12;\pm}(\lambda)$ are regular at $\lambda = 0$. In their turn, $\chi_{11}, \chi_{12}$ admit meromorphic continuations from $\mathbb{H}^{\pm}$ into some small tubular neighbourhood of $\mathbb{R}$ in $\mathbb{H}^{\pm}$. They admit only one pole, which is simple at $\lambda = 0$, and one has the behaviour

$$ \chi_{21;-(\lambda)} = R_{\downarrow}(0) \tilde{\vartheta}_R \Pi_{11}(0) \cdot \frac{1}{\lambda} + O(1) \quad \text{and} \quad \chi_{22;-(\lambda)} = R_{\downarrow}(0) \tilde{\vartheta}_R \Pi_{12}(0) \cdot \frac{1}{\lambda} + O(1) \qquad (2.38) $$

as $\lambda \to 0^+$.

### $\lambda$ between $\mathbb{R} + i\epsilon$ and $\Gamma_{\uparrow}$

Here, $\epsilon > 0$ is some fixed, small enough, constant. It can be taken so that $\epsilon' > \epsilon$, with $\epsilon'$ as appearing in (2.23). In this region, it holds that

$$ \chi(\lambda) = \chi_{\infty}(\lambda) + \delta\chi(\lambda) \qquad (2.39) $$

where

$$ \chi_{\infty}(\lambda) = \begin{pmatrix} 1/[\lambda R_{\uparrow}(\lambda)] - e^{i\lambda \bar{x}_{N;\alpha}}/R_{\downarrow}(\lambda) & 1/R_{\uparrow}(\lambda) \\ -R_{\uparrow}(\lambda) & 0 \end{pmatrix}, \qquad (2.40) $$

while

$$ \delta\chi(\lambda) = \begin{pmatrix} \left\{ \frac{1}{\lambda R_{\uparrow}(\lambda)} - \frac{e^{i\lambda \bar{x}_{N;\alpha}}}{R_{\downarrow}(\lambda)} \right\} \cdot [\delta(\Pi_{PR})(\lambda)]_{11} + \frac{[\delta(\Pi_{PR})(\lambda)]_{21}}{R_{\uparrow}(\lambda)} & -R_{\uparrow}(\lambda)[\delta(\Pi_{PR})(\lambda)]_{11} \\ \frac{1}{\lambda R_{\uparrow}(\lambda)} - \frac{e^{i\lambda \bar{x}_{N;\alpha}}}{R_{\downarrow}(\lambda)} \cdot [\delta(\Pi_{PR})(\lambda)]_{12} + \frac{[\delta(\Pi_{PR})(\lambda)]_{22}}{R_{\uparrow}(\lambda)} & -R_{\uparrow}(\lambda)[\delta(\Pi_{PR})(\lambda)]_{12} \end{pmatrix}. \qquad (2.41) $$

The formulae for the remainder matrix involve

$$ \delta(\Pi_{PR})(\lambda) = \begin{pmatrix} 1 & 0 \\ -1/\lambda & 1 \end{pmatrix} \cdot \left[ \Pi_{PR}(\lambda) - \begin{pmatrix} 1 & 0 \\ 1/\lambda & 1 \end{pmatrix} \right]. \qquad (2.42) $$

Finally, one has the uniform entrywise estimate on the remainder

$$ \delta\chi(\lambda) = O(e^{-\zeta(1-\eta)\bar{x}_N}). \qquad (2.43) $$

### $\lambda$ between $\Gamma_{\downarrow}$ and $\mathbb{R}$

In this region, it holds that

$$ \chi(\lambda) = \chi_{\infty}(\lambda) + \delta\chi(\lambda) \qquad (2.44) $$

where

$$ \chi_{\infty}(\lambda) = \begin{pmatrix} \frac{-1}{R_{\downarrow}(\lambda)} \left( 1 - \frac{R_{\downarrow}(\lambda)}{\lambda R_{\uparrow}(\lambda)} e^{-i\lambda \bar{x}_{N;\alpha}} \right) & \frac{e^{-i\lambda \bar{x}_{N;\alpha}}}{R_{\uparrow}(\lambda)} \\ \frac{R_{\downarrow}(\lambda)}{\lambda} & R_{\downarrow}(\lambda) \end{pmatrix}, \qquad (2.45) $$
<figure><img src="image_2.png" /><figcaption>Figure 1: Contour $\Gamma_{\uparrow/\downarrow}$ delimiting regions of uniform asymptotic expansion of $\chi$.</figcaption></figure>

while

$$ \delta\chi(\lambda) = \left( \begin{array}{l} \frac{-1}{R_{\downarrow}(\lambda)} \left(1 - \frac{R_{\downarrow}(\lambda)}{\lambda R_{\uparrow}(\lambda)} e^{-i\lambda \bar{x}_{N;\alpha}}\right) \cdot [\delta(\Pi_{P_R})(\lambda)]_{11} + \frac{e^{-i\lambda \bar{x}_{N;\alpha}}}{R_{\uparrow}(\lambda)} \cdot [\delta(\Pi_{P_R})(\lambda)]_{21} \\ \frac{R_{\downarrow}(\lambda)}{\lambda} \cdot [\delta(\Pi_{P_R})(\lambda)]_{11} + R_{\downarrow}(\lambda)[\delta(\Pi_{P_R})(\lambda)]_{21} \\ \frac{-1}{R_{\downarrow}(\lambda)} \left(1 - \frac{R_{\downarrow}(\lambda)}{\lambda R_{\uparrow}(\lambda)} e^{-i\lambda \bar{x}_{N;\alpha}}\right) \cdot [\delta(\Pi_{P_R})(\lambda)]_{12} + \frac{e^{-i\lambda \bar{x}_{N;\alpha}}}{R_{\uparrow}(\lambda)} \cdot [\delta(\Pi_{P_R})(\lambda)]_{22} \\ \frac{R_{\downarrow}(\lambda)}{\lambda} \cdot [\delta(\Pi_{P_R})(\lambda)]_{12} + R_{\downarrow}(\lambda)[\delta(\Pi_{P_R})(\lambda)]_{22} \end{array} \right). \quad (2.46) $$

One has the uniform entrywise estimate on the remainder

$$ \delta\chi(\lambda) = O(e^{-\zeta(1-\eta)\bar{x}_N}). \quad (2.47) $$

Moreover, one infers that

$$ \chi_{21}(\lambda) = \frac{R_{\downarrow}(0) \bar{\vartheta}_R \Pi_{11}(0)}{\lambda} + O(1) \quad \text{as} \quad \lambda \to 0 \quad \text{with} \quad \mathrm{Im}(\lambda) < 0. \quad (2.48) $$

$$ \chi_{22}(\lambda) = \frac{R_{\downarrow}(0) \bar{\vartheta}_R \Pi_{12}(0)}{\lambda} + O(1) $$

and that $\chi_{11}(\lambda)$ and $\chi_{12}(\lambda)$ admit bounded limits as $\lambda \to 0$.

A remark is in order: while both the subspace constraint functional $\mathcal{J}$ and the inverse $\mathcal{W}_N$ involve the Fourier transform of $g$, their values only depend on the values of $g$ on $[a_N; b_N]$. This property basically follows from the jump conditions of $\chi$. It will allow us, later on, to simplify some of the handlings.

**Lemma 2.3.** Let $g_1, g_2 \in H_s(\mathbb{R})$, $0 < s < 1/2$ be such that $g_1 = g_2$ on $[a_N; b_N]$. Then,

$$ \mathcal{J}[g_1] = \mathcal{J}[g_2] \quad \text{and} \quad \mathcal{W}_N[g_1] = \mathcal{W}_N[g_2]. \quad (2.49) $$
Proof—

Since $g\mathbf{1}_{[a_N; b_N]} \in H_s(\mathbb{R})$, $0 < s < 1/2$ as soon as $g \in H_s(\mathbb{R})$, it is enough to show that $\mathcal{J}[g]$ and $\mathcal{W}_N[g]$ only depend on $g\mathbf{1}_{[a_N; b_N]}$. We discuss the proof only in the case of sufficiently regular $g$ having fast decay at infinity although this can be done in full generality within the distributional setting of $H_s(\mathbb{R})$ functions.

One may decompose $\mathcal{J}$ in the form

$$ \mathcal{J}[g] = \mathcal{J}_L[g] + \mathcal{J}_c[g] + \mathcal{J}_R[g] \qquad (2.50) $$

with

$$ \mathcal{J}_L[g] = \int_{\mathbb{R}+i\epsilon'} \frac{d\mu}{2i\pi} \chi_{11}(\mu) \int_{-\infty}^{a_N} d\eta g(\eta) e^{i\tau_N \mu(\eta-b_N)}, \quad \mathcal{J}_R[g] = \int_{\mathbb{R}+i\epsilon'} \frac{d\mu}{2i\pi} \chi_{11}(\mu) \int_{b_N}^{+\infty} d\eta g(\eta) e^{i\tau_N \mu(\eta-b_N)} \qquad (2.51) $$

and $\mathcal{J}_c[g] = \mathcal{J}[g\mathbf{1}_{[a_N; b_N]}]$.

We now show the vanishing of $\mathcal{J}_L[g]$ and $\mathcal{J}_R[g]$. Indeed, by using that $\chi_{11}$ is analytic in the upper half plane, that

$$ \chi_{11}(\mu) \le \frac{C}{|\mu|^{1/2}} \quad \text{and} \quad \int_{b_N}^{+\infty} d\eta g(\eta) e^{i\tau_N \mu(\eta-b_N)} = \frac{-g(b_N)}{i\mu\tau_N} - \frac{1}{i\mu\tau_N} \int_{b_N}^{+\infty} d\eta g'(\eta) e^{i\tau_N \mu(\eta-b_N)}, \qquad (2.52) $$

one observes that one has the uniform bound on $\mathbb{H}^+$

$$ \left| \chi_{11}(\mu) \int_{b_N}^{+\infty} d\eta g(\eta) e^{i\tau_N \mu(\eta-b_N)} \right| \le \frac{C}{|\mu|^{3/2}} \qquad (2.53) $$

and that the bounded functions appearing above are analytic in $\mathbb{H}^+$. This allows one to deform the integrations from $\mathbb{R} + i\epsilon$ to $\mathbb{R} + iM$ with $M > 0$ as large as desired by virtue of Morera's theorem. Then, one has that

$$ |\mathcal{J}_R[g]| = \left| \int_{\mathbb{R}+iM} \frac{d\mu}{2i\pi} \chi_{11}(\mu) \int_{b_N}^{+\infty} d\eta g(\eta) e^{i\tau_N \mu(\eta-b_N)} \right| \le C \int_{\mathbb{R}} \frac{ds}{[s^2 + M^2]^{3/2}} = \frac{C}{\sqrt{M}} \int_{\mathbb{R}} \frac{ds}{[s^2 + 1]^{3/2}} \underset{M \to +\infty}{\longrightarrow} 0. \qquad (2.54) $$

One carries out a similar reasoning regarding to $\mathcal{J}_L[g]$. The $\mu^{-1}$ decay rate of the Fourier transform at infinity and the existence of continuous + boundary values of $\chi_{11}$ on $\mathbb{R}$ ensure that

$$ \begin{aligned} \mathcal{J}_L[g] &= \int_{\mathbb{R}} \frac{d\mu}{2i\pi} \chi_{11;+}(\mu) \int_{-\infty}^{a_N} d\eta g(\eta) e^{i\tau_N \mu(\eta-b_N)} \\ &= \int_{\mathbb{R}} \frac{d\mu}{2i\pi} \chi_{11;-}(\mu) \int_{-\infty}^{a_N} d\eta g(\eta) e^{-i\tau_N \mu(a_N-\eta)} \\ &= \int_{\mathbb{R}-iM} \frac{d\mu}{2i\pi} \chi_{11}(\mu) \int_{-\infty}^{a_N} d\eta g(\eta) e^{-i\tau_N \mu(a_N-\eta)} \underset{M \to +\infty}{\longrightarrow} 0. \end{aligned} \qquad (2.55) $$

This entails the claim relative to $\mathcal{J}[g]$. The results relative to $\mathcal{W}_N[g]$ are obtained in a very similar fashion. ■
## 2.3 A convenient representation for the inverse acting on $V'_{N;\alpha}$

In the general case, one may not expect to be able to simplify $W_N[g]$ beyond its two-dimensional integral representation. However, since the potential of interest to us takes a very simple and specific form, such simplifications are possible in the case of $W_N[V'_{N;\alpha}]$.

We first observe that the expression (2.39) for $\chi$ on the line $\mathbb{R} + i\epsilon'$, $\epsilon' > \epsilon$ leads to the representation

$$ \chi_{11}(\lambda) = \frac{Q_{\uparrow}(\lambda)}{\lambda R_{\uparrow}(\lambda)} - e^{i\lambda \bar{x}_N} \frac{Q_{\downarrow}(\lambda)}{R_{\downarrow}(\lambda)} \quad \text{with} \quad \left\{ \begin{array}{l} Q_{\uparrow}(\lambda) = 1 + \delta(\Pi_{P_R})_{11}(\lambda) + \lambda \delta(\Pi_{P_R})_{21}(\lambda) \\ Q_{\downarrow}(\lambda) = 1 + \delta(\Pi_{P_R})_{11}(\lambda) \end{array} \right. . \quad (2.56) $$

There, the functions $Q_{\uparrow/\downarrow}$ are holomorphic and bounded in the region enclosed by the curves $\Gamma_{\uparrow}$ and $\Gamma_{\downarrow}$. Similarly,

$$ \chi_{12}(\lambda) = \frac{\tilde{Q}_{\uparrow}(\lambda)}{\lambda R_{\uparrow}(\lambda)} - e^{i\lambda \bar{x}_N} \frac{\tilde{Q}_{\downarrow}(\lambda)}{R_{\downarrow}(\lambda)} \quad \text{with} \quad \left\{ \begin{array}{l} \tilde{Q}_{\uparrow}(\lambda) = \lambda + \delta(\Pi_{P_R})_{12}(\lambda) + \lambda \delta(\Pi_{P_R})_{22}(\lambda) \\ \tilde{Q}_{\downarrow}(\lambda) = \delta(\Pi_{P_R})_{12}(\lambda) \end{array} \right. . \quad (2.57) $$

For further purpose, it is convenient to introduce the vectors

$$ E_L(\lambda) = \begin{pmatrix} \chi_{11}(\lambda) \\ -\frac{\chi_{12}(\lambda)}{\lambda} \end{pmatrix}, \quad E_R(\lambda) = \begin{pmatrix} \chi_{12}(\lambda) \\ \lambda \chi_{11}(\lambda) \end{pmatrix} \quad (2.58) $$

so that

$$ (E_L(\lambda), E_R(\mu)) = \chi_{11}(\lambda)\chi_{12}(\mu) - \frac{\mu}{\lambda}\chi_{11}(\mu)\chi_{12}(\lambda). \quad (2.59) $$

The decompositions (2.56)-(2.57) for $\chi_{11}$ and $\chi_{12}$ entail that $E_R(\lambda) = E_R^{(\uparrow)}(\lambda) - e^{i\lambda \bar{x}_N} E_R^{(\downarrow)}(\lambda)$ with

$$ E_R^{(\uparrow)}(\lambda) = \frac{1}{\lambda R_{\uparrow}(\lambda)} \begin{pmatrix} \tilde{Q}_{\uparrow}(\lambda) \\ \lambda Q_{\uparrow}(\lambda) \end{pmatrix}, \quad E_R^{(\downarrow)}(\lambda) = \frac{1}{R_{\downarrow}(\lambda)} \begin{pmatrix} \tilde{Q}_{\downarrow}(\lambda) \\ \lambda Q_{\downarrow}(\lambda) \end{pmatrix}, \quad (2.60) $$

and similarly $E_L(\lambda) = \frac{1}{\lambda} \cdot E_L^{(\uparrow)}(\lambda) - \frac{e^{i\lambda \bar{x}_N}}{\lambda} \cdot E_L^{(\downarrow)}(\lambda)$

$$ E_L^{(\uparrow)}(\lambda) = \frac{1}{\lambda R_{\uparrow}(\lambda)} \begin{pmatrix} \lambda Q_{\uparrow}(\lambda) \\ -\tilde{Q}_{\uparrow}(\lambda) \end{pmatrix}, \quad E_L^{(\down)}(\lambda) = \frac{1}{R_{\downarrow}(\lambda)} \begin{pmatrix} \lambda Q_{\downarrow}(\lambda) \\ -\tilde{Q}_{\downarrow}(\lambda) \end{pmatrix}. \quad (2.61) $$

Finally, we also set

$$ \mathcal{W}_{12}(\lambda) = \frac{Nu_N\lambda + i\nu_N}{1 + \lambda^2} i\chi_{11}(i), \quad (2.62) $$

$$ \mathcal{W}_{11}(\lambda) = \frac{-\chi_{12}(i)}{1 + \lambda^2} \cdot (iNu_N + \lambda\nu_N) - \frac{i\chi_{11}(i)Nu_N - \lambda\nu_N}{i + \lambda} \cdot \frac{2}{2}, \quad (2.63) $$

in which we have introduced the shorthand notations

$$ Nu_N = e^{\bar{b}_N} + e^{-\bar{a}_N} \quad \text{and} \quad \nu_N = e^{\bar{b}_N} - e^{-\bar{a}_N}. \quad (2.64) $$

**Proposition 2.4.** One has the decomposition $W_N[V'_{N;\alpha}] = \sum_{a=1}^{3} \omega_N^{(a)}$ where

$$ \omega_N^{(1)}(\xi) = \frac{r\tau_N}{4i\pi N} \int_{\mathbb{R}+2i\epsilon'} \frac{d\lambda}{2i\pi} e^{-i\tau_N\lambda(\xi-a_N)} \{\frac{\chi_{12}(\lambda)}{\lambda} \mathcal{W}_{12}(\lambda) + \chi_{11}(\lambda)\mathcal{W}_{11}(\lambda)\}, \quad (2.65) $$
and

$$ \bar{\omega}_N^{(2)}(\xi) = \frac{\alpha \tau_N}{2i\pi N} \int_{\mathbb{R}+2i\epsilon'} \frac{d\lambda}{2i\pi} e^{-i\tau_N \lambda (\xi-a_N)} \frac{\chi_{11}(\lambda)\chi_{12};-(0)}{\lambda} . \qquad (2.66) $$

Finally, it holds that $\bar{\omega}_N^{(3)}(\xi) = \bar{\omega}_{N;\uparrow}^{(3)}(\xi) + \bar{\omega}_{N;\downarrow}^{(3)}(\xi) + \bar{\omega}_{N;0}^{(3)}(\xi)$. The building blocks of this decomposition take the form, for $\nu \in \{\uparrow, \downarrow\}$,

$$ \bar{\omega}_{N;\nu}^{(3)}(\xi) = \frac{\epsilon_{\nu} \tau_N}{4i\pi N} \int_{\mathbb{R}+2i\epsilon'} \frac{d\lambda}{2i\pi} e^{-i\tau_N \lambda (\xi-a_N)} (\mathbf{E}_L(\lambda), \mathcal{E}^{(\nu)}(\lambda)) \quad \text{with} \quad \epsilon_{\nu} = \begin{cases} 1 & \nu = \uparrow \\ -1 & \nu = \downarrow \end{cases} . \qquad (2.67) $$

Here, we have set

$$ \mathcal{E}^{(\uparrow)}(\lambda) = e^{-\bar{b}_N \frac{2\mathcal{F}[g](i)}{\pi(\lambda+i)}} \mathbf{E}_R^{(\uparrow)}(-i)\mathbf{1}_{1<\zeta} + \int_{\mathbb{R}-i\kappa_\eta} \frac{d\mu}{2i\pi} \frac{\mu\mathcal{F}[g](\mu) e^{-i\bar{b}_N\mu}}{(\mu-\lambda)\cosh\left[\frac{\pi\mu}{2}\right]} \mathbf{E}_R^{(\uparrow)}(\mu) \qquad (2.68) $$

$$ \mathcal{E}^{(\downarrow)}(\lambda) = e^{\bar{a}_N \frac{2\mathcal{F}[g](i)}{\pi(i-\lambda)}} \mathbf{E}_R^{(\downarrow)}(i)\mathbf{1}_{1<\zeta} + \int_{\mathbb{R}+i\kappa_\eta} \frac{d\mu}{2i\pi} \frac{\mu\mathcal{F}[g](\mu) e^{-i\bar{a}_N\mu}}{(\mu-\lambda)\cosh\left[\frac{\pi\mu}{2}\right]} \mathbf{E}_R^{(\downarrow)}(\mu) \qquad (2.69) $$

while

$$ \bar{\omega}_{N;0}^{(3)}(\xi) = \frac{-\tau_N}{4i\pi N} \int_{\mathbb{R}+2i\epsilon'} \frac{d\lambda}{2i\pi} e^{-i\tau_N \lambda \xi} (\mathbf{E}_L(\lambda), \mathbf{E}_R^{(\downarrow)}(\lambda)) \frac{\lambda \mathcal{F}[g](\lambda)}{\cosh\left[\frac{\pi \lambda}{2}\right]} . \qquad (2.70) $$

Above, we have introduced

$$ \kappa_{\eta} = (1 - \eta) \min\{2, \zeta\} \quad \text{with} \quad \zeta = 2\pi \frac{\omega_1 \omega_2}{\omega_1 + \omega_2} . \qquad (2.71) $$

Proof—

Owing to Lemma 2.3, one may choose $V'_{N;\alpha}$ to take any values outside of $[a_N; b_N]$ so as to compute the Fourier transform occurring in the expression for $W_N$, provided the function belongs to $H_s(\mathbb{R})$ with $0 < s < 1/2$. Thus, we choose to extend $V'_{N;\alpha}$ from $[a_N; b_N]$ to $\mathbb{R}$ as $V_{N;\alpha}(\lambda) = \nu_{N;\alpha}(\lambda)\mathbf{1}_{[a_N; b_N]}(\lambda) + w_N(\lambda)$, where

$$ \nu_{N;\alpha}(\lambda) = \frac{r}{N\tau_N} \cosh[\tau_N\lambda] - \frac{\alpha\lambda}{N} \quad \text{and} \quad w_N(\lambda) = \int_{\mathbb{R}} \frac{d\mu}{2\pi N} \cdot \frac{g(\tau_N\mu)}{\cosh[\tau_N(\lambda - \mu)]} . \qquad (2.72) $$

A direct calculation yields

$$ \int_{a_N}^{b_N} ds \, e^{i\mu\tau_N(s-b_N)} v'_{N;\alpha}(s) = \frac{r}{2\tau_N N i} \sum_{\sigma=\pm} \sigma \frac{e^{\sigma\bar{b}_N} - e^{-i\mu\bar{x}_N} e^{\sigma\bar{a}_N}}{\mu - i\sigma} - \alpha \frac{1 - e^{-i\mu\bar{x}_N}}{i\mu N \tau_N} . \qquad (2.73) $$

Further, one has that

$$ \begin{aligned} \int_{\mathbb{R}} ds \, e^{i\mu\tau_N s} w'_{N}(s) &= -i\mu\tau_N \int_{\mathbb{R}} ds \, w_N(s) e^{i\mu\tau_N s} \\ &= -\frac{i\mu\tau_N}{2\pi N} \int_{\mathbb{R}} ds \frac{e^{i\mu\tau_N s}}{\cosh(\tau_N s)} \cdot \int_{\mathbb{R}} dt \, e^{i\mu\tau_N t} g(\tau_N t) = -\frac{i\mu \mathcal{F}[g](\mu)}{2\tau_N N \cosh\left[\frac{\pi\mu}{2}\right]} . \end{aligned} \qquad (2.74) $$
As a result, one obtains a decomposition $\mathcal{W}_N[V'_{N;\alpha}] = \sum_{a=1}^3 \varpi_{N;\alpha}^{(a)}$ where

$$ \varpi_N^{(1)}(\xi) = \frac{\tau_N r}{4i\pi N} \int_{\mathbb{R}+2i\epsilon'} \frac{d\lambda}{2i\pi} \int_{\mathbb{R}+i\epsilon'} \frac{d\mu}{2i\pi} \frac{e^{-i\tau_N \lambda(\xi-a_N)}}{\mu-\lambda} \{\chi_{11}(\lambda)\chi_{12}(\mu) - \frac{\mu}{\lambda}\chi_{11}(\mu)\chi_{12}(\lambda)\} \sum_{\sigma=\pm} \sigma \frac{e^{\sigma\bar{b}_N} - e^{-i\mu\bar{\chi}_N}e^{\sigma\bar{a}_N}}{\mu - i\sigma}, $$

$$ \varpi_N^{(2)}(\xi) = -\frac{\tau_N \alpha}{2i\pi N} \int_{\mathbb{R}+2i\epsilon'} \frac{d\lambda}{2i\pi} \int_{\mathbb{R}+i\epsilon'} \frac{d\mu}{2i\pi} \frac{e^{-i\tau_N \lambda(\xi-a_N)}}{\mu-\lambda} \{\chi_{11}(\lambda)\chi_{12}(\mu) - \frac{\mu}{\lambda}\chi_{11}(\mu)\chi_{12}(\lambda)\} \frac{1-e^{-i\mu\bar{\chi}_N}}{\mu}, $$

and

$$ \varpi_N^{(3)}(\xi) = -\frac{i\tau_N}{4\pi N} \int_{\mathbb{R}+2i\epsilon'} \frac{d\lambda}{2i\pi} \int_{\mathbb{R}+i\epsilon'} \frac{d\mu}{2i\pi} \frac{e^{-i\tau_N \lambda(\xi-a_N)}}{\mu-\lambda} \{\chi_{11}(\lambda)\chi_{12}(\mu) - \frac{\mu}{\lambda}\chi_{11}(\mu)\chi_{12}(\lambda)\} e^{-i\mu\bar{b}_N} \frac{\mu \mathcal{F}[g](\mu)}{\cosh\left[\frac{\pi\mu}{2}\right]}. \quad (2.75) $$

We first compute the $\mu$-integral arising in $\varpi_N^{(1)}$ and $\varpi_N^{(2)}$. For such a purpose, one starts by observing that $\chi_{1a}$ admits an analytic continuation from $\mathbb{H}^-$ to $\mathbb{H}^+$. Denoting this analytic continuation as $\chi_{1a,-}$, one has the relation

$$ \chi_{1a,-}(\lambda) = \chi_{1a}(\lambda)e^{-i\lambda\bar{\chi}_N} \quad \text{with} \quad \mathfrak{F}[\lambda] > 0. \qquad (2.76) $$

Then, one may express $\varpi_N^{(1)}$ in the form

$$ \begin{aligned} \varpi_N^{(1)}(\xi) = & \frac{\tau_N r}{4i\pi N} \int_{\mathbb{R}+2i\epsilon'} \frac{d\lambda}{2i\pi} \int_{\mathbb{R}+i\epsilon'} \frac{d\mu}{2i\pi} \frac{e^{-i\tau_N \lambda(\xi-a_N)}}{\mu-\lambda} \{\chi_{11}(\lambda)\chi_{12}(\mu) - \frac{\mu}{\lambda}\chi_{11}(\mu)\chi_{12}(\lambda)\} \sum_{\sigma=\pm} \frac{\sigma e^{\sigma\bar{b}_N}}{\mu - i\sigma} \\ & - \frac{\tau_N r}{4i\pi N} \int_{\mathbb{R}+2i\epsilon'} \frac{d\lambda}{2i\pi} \int_{\mathbb{R}+i\epsilon'} \frac{d\mu}{2i\pi} \frac{e^{-i\tau_N \lambda(\xi-a_N)}}{\mu-\lambda} \{\chi_{11}(\lambda)\chi_{12;,-}(\mu) - \frac{\mu}{\lambda}\chi_{11;,-}(\mu)\chi_{12}(\lambda)\} \sum_{\sigma=\pm} \frac{\sigma e^{\sigma\bar{a}_N}}{\mu - i\sigma} \end{aligned} \quad (2.77) $$

We could split the integral in two pieces since each integrand behaves at infinity as $O(|\mu|^{-3/2})$. Then, because of these bounds, one may take the first $\mu$-integral by means of the residues of the poles located above the line $\mathbb{R}+i\epsilon'$ and take the second $\mu$-integral by means of the residues of the poles located below the line $\mathbb{R}+i\epsilon'$. Note that there is no pole at $\mu - \lambda$ in the first integral so that only the pole at $\mu = i$ contributes, while, in the second case, only the pole at $\mu = -i$ does. This yields

$$ \varpi_N^{(1)}(\xi) = \frac{r\tau_N}{4i\pi N} \int_{\mathbb{R}+2i\epsilon'} \frac{d\lambda}{2i\pi} e^{-i\tau_N \lambda(\xi-a_N)} \sum_{\sigma=\pm} \frac{e^{\bar{b}_N^{(\sigma)} \chi_{11}(\lambda)\chi_{12}(\sigma i) - \sigma i \chi_{11}(\sigma i)\chi_{12}(\lambda)/\lambda}}{i - \sigma i\lambda} \quad (2.78) $$

in which we have used the shorthand notations

$$ \bar{b}_N^{(+)} = \bar{b}_N \quad \text{and} \quad \bar{b}_N^{(-)} = -\bar{a}_N. \qquad (2.79) $$

Then, it is a matter of direct calculation to observe that owing to the inversion relation (2.26) one gets

$$ \sum_{\sigma=\pm} e^{\bar{b}_{N;\sigma}^{(\sigma)}} \frac{\chi_{11}(\lambda)\chi_{12}(\sigma i) - \sigma i\chi_{11}(\sigma i)\chi_{12}(\lambda)/\lambda}{i - \sigma i\lambda} = \frac{\chi_{12}(\lambda)}{\lambda} \mathcal{W}_{12}(\lambda) + \chi_{11}(\lambda) \mathcal{W}_{11}(\lambda) \quad (2.80) $$

with $\mathcal{W}_{1a}$ as given in (2.62)-(2.63). This was the last step before reaching (2.65).
The same reasoning yields (2.66) starting from the previous expression for $\omega_N^{(2)}$: the part of the integral deformed up to $+i\infty$ produces 0 while the part deformed to $-i\infty$ picks a simple pole at $\mu = 0$.

We now turn on to rewriting $\omega_N^{(3)}$, which can be recast as

$$ \omega_N^{(3)}(\xi) = \frac{\tau_N}{4i\pi N} \int_{\mathbb{R}+2i\epsilon'} \frac{d\lambda}{2i\pi} \int_{\mathbb{R}+i\epsilon'} \frac{d\mu}{2i\pi} \frac{e^{-i\tau_N\lambda(\xi-a_N)}}{\mu-\lambda} \left\{ e^{-i\mu\bar{b}_N} (E_L(\lambda), E_R^{(\uparrow)}(\mu)) - e^{-i\mu\bar{a}_N} (E_L(\lambda), E_R^{(\downarrow)}(\mu)) \right\} \frac{\mu \mathcal{F}[g](\mu)}{\cosh\left[\frac{\pi\mu}{2}\right]} . \quad (2.81) $$

One then splits the integral in two pieces, one containing $E_R^{(\uparrow)}$ and the other one $E_R^{(\downarrow)}$. Both integrals are well defined due to the $O(\mu^{-k})$ behaviour at infinity of $\mathcal{F}[g](\mu)$ ensured by the $O(e^{-\frac{\epsilon}{2}\cosh(\xi)})$ control on $g^{(k)}(\xi)$ for any $k \ge 0$ and the $O(|\lambda|^{-3/2})$ behaviour of the integrand pointwise in $\mu$. Then, in the integral involving $E_R^{(\uparrow)}$, one moves the $\mu$ integration from $\mathbb{R}+i\epsilon'$ to $\mathbb{R}-i\kappa_\eta$. There are four potential sources of poles in the integrand

$$ \frac{1}{R_{\uparrow}(\mu)} \quad \text{poles at } \mu = -i\zeta n, \quad \frac{1}{R_{\downarrow}(\mu)} \quad \text{poles at } \mu = i\zeta n, \quad \frac{1}{\cosh[\frac{\pi\mu}{2}]} \quad \text{poles at } \mu = \pm i(1+2n) \quad (2.82) $$

with $n \in \mathbb{N}^*$, and $\mu = \lambda$. Thus, in deforming the integration contour, provided that $1 < \zeta$, one picks up a pole at $\mu = -i$, else no poles are crossed. This then yields $\omega_{N;\uparrow}^{(3)}(\xi)$

Similarly, in the second integral involving $E_R^{(\downarrow)}$, one moves the $\mu$-integration contour from $\mathbb{R}+i\epsilon'$ to $\mathbb{R}+i\kappa_\eta$. This produces one contribution stemming from the pole at $\mu = \lambda$ and one contribution stemming from the pole at $\mu = i$. The last contribution is only present if $1 < \zeta$. The terms obtained in this way correspond to $\omega_{N,\downarrow}^{(3)}(\xi)$ for the $\mu$-integrals over $\mathbb{R}+i\kappa_\eta$ and $\omega_{N,0}^{(3)}(\xi)$ for the residue at $\mu = \lambda$ part. ■

## 2.4 Support of the equilibrium measure

When constructing the equilibrium measure, on top of determining its density, one also needs to fix its support. Since the density belongs to $H_s(\mathbb{R})$, see [7], with $0 < s < 1/2$, and satisfies the singular integral equation (2.16) throughout its support $[a_{N;\alpha}; b_{N;\alpha}]$ which satisfies owing to Lemma 2.2 the lower-bound $b_{N;\alpha} - a_{N;\alpha} \ge 2\zeta > 0$, one gets that, for any $N$ large enough, one has the representation

$$ \widehat{\mathcal{Q}}_{\text{eq};\alpha} = \mathcal{W}_N[V'_{N;\alpha}]|_{a_N,b_N \leftrightarrow a_{N;\alpha},b_{N;\alpha}}. \quad (2.83) $$

We stress that the inverse operator $\mathcal{W}_N$ given in (2.23) is now subordinate to the yet unknown pair of points $a_{N;\alpha}, b_{N;\alpha}$ delimiting the support. One then gets two additional constraints, the first one translating the fact that $V'_{N;\alpha} \in S_N[H_s([a_{N;\alpha}; b_{N;\alpha}])]$ with $0 < s < 1/2$, and the second one expressing the unit mass property of the measure

$$ 0 = \mathcal{J}[V'_{N;\alpha}]|_{a_N,b_N \leftrightarrow a_{N;\alpha},b_{N;\alpha}} \quad \text{and} \quad 1 = \int_{a_{N;\alpha}}^{b_{N;\alpha}} d\xi \{\mathcal{W}_N[V'_{N;\alpha}](\xi)\}|_{a_N,b_N \leftrightarrow a_{N;\alpha},b_{N;\alpha}}. \quad (2.84) $$

In this subsection, we shall establish that the constraints (2.84) admit a unique solution, for $N$ large enough, provided that $a_{N;\alpha} \le -\zeta$ and $b_{N;\alpha} \ge \zeta$, a property that is ensured by Lemma 2.2. Hence, this solution does provide one with the support of the equilibrium measure. We close the subsection by establishing the explicit form of the first few terms in the large-$N$ expansion of $a_{N;\alpha}$ and $b_{N;\alpha}$. This ends the proof of Theorem 1.1.

**Proposition 2.5.** For given endpoints $a_N, b_N$ satisfying $x_N > \eta$ for some $\eta > 0$, the constraint functional $\mathcal{I}_1[V'_{N;\alpha}]$ defined in (2.21) admits the large-$N$ asymptotic expansion

$$ \mathcal{J}[V'_{N;\alpha}] = \frac{r\chi_{11}(i)}{2iN\tau_N}(e^{\bar{b}_N} - e^{-\bar{a}_N}) - \frac{\alpha\chi_{11;-(0)}}{iN\tau_N} - \frac{\mathcal{F}[g](i)}{\pi N \tau_N R_{\uparrow}(-i)}\{e^{-\bar{b}_N} - e^{-\bar{a}_N}\} \mathbf{1}_{1<\zeta} + \mathfrak{R}\mathcal{J}(a_N, b_N) \quad (2.85) $$
with a remainder that is smooth in $a_N, b_N$ and controlled as

$$ \partial_{a_N}^k \partial_{b_N}^l \Re \mathcal{J}(a_N, b_N) = O\left(\tau_N^{k+l} \frac{e^{-\bar{b}_N \kappa_\eta} + e^{-\bar{a}_N \kappa_\eta}}{N \tau_N}\right). \qquad (2.86) $$

$\kappa_\eta$ appearing above is as introduced in (2.71).

Proof—

One consecutively computes each of the contributions to $\mathcal{J}$. The one of $v_{N;\alpha}$ can be obtained in closed form. Indeed, one has

$$ \mathcal{J}[v'_{N;\alpha}] = \int_{\mathbb{R}+i\epsilon'} \frac{d\mu}{2i\pi N \tau_N} \chi_{11}(\mu) \left\{ \sum_{\sigma=\pm} \frac{r\sigma e^{\sigma\bar{b}_N}}{2i(\mu - i\sigma)} - \frac{\alpha}{i\mu} \right\} \\ - \int_{\mathbb{R}+i\epsilon'} \frac{d\mu}{2i\pi \tau_N N} \chi_{11}(\mu) e^{-i\mu\bar{x}_N} \left\{ \sum_{\sigma=\pm} \frac{r\sigma e^{\sigma\bar{a}_N}}{2i(\mu - i\sigma)} - \frac{\alpha}{i\mu} \right\}. \quad (2.87) $$

Note that, each integrand is a $O(\mu^{-3/2})$ at $\infty$. The first integral can be computed by taking the residues of the poles located above of $\mathbb{R} + i\epsilon'$. There is a simple pole at $\mu = i$. To compute the second integral, one observes that $\chi_{11}$ admits an analytic continuation from $\mathbb{H}^-$ to $\mathbb{H}^+$. Denoting this analytic continuation as $\chi_{11;-,}$, it holds $\chi_{11;-(\lambda)} = \chi_{11}(\lambda)e^{-i\lambda\bar{x}_N}$ with $\Re[\lambda] > 0$. Thus, in the second integral, one replaces $\chi_{11}$ with $\chi_{11;-,}$ and then takes the integral in terms of the residues at the poles located below of $\mathbb{R} + i\epsilon'$. There are two poles, one simple at $\mu = i$ and one simple at $\mu = 0$. This yields

$$ \mathcal{J}[v'_{N;\alpha}] = \frac{r}{2i\tau_N N} (\chi_{11}(i)e^{\bar{b}_N} - \chi_{11}(-i)e^{-\bar{a}_N}) - \frac{\alpha\chi_{11;-(0)}}{i\tau_N N} \quad (2.88) $$

$$ = \frac{r\chi_{11}(i)}{2i\tau_N N} (e^{\bar{b}_N} - e^{-\bar{a}_N}) - \frac{\alpha\chi_{11;-(0)}}{i\tau_N N}. \qquad (2.89) $$

Here, we have simplified the expression owing to (2.27).

Further, observe that integrations by parts and the $O(e^{-r\cosh(\xi)})$ decay of $g^{(k)}(\xi)$ for any $k \ge 0$ ensure that $\mathcal{F}[g](\mu) = O(\mu^{-k})$. Then, recalling the representations (2.56), (2.74) and inserting into the expression for the constraint functional leads to

$$ \begin{aligned} \mathcal{J}[w'_{N}] &= \int_{\mathbb{R}+i\epsilon'} \frac{d\mu}{4i^2\pi N \tau_N} \frac{Q_{\uparrow}(\mu)\mathcal{F}[g](\mu)}{R_{\uparrow}(\mu)\cosh\left[\frac{\pi\mu}{2}\right]} e^{-i\mu\bar{b}_N} - \int_{\mathbb{R}+i\epsilon'} \frac{d\mu}{4i^2\pi N \tau_N} \frac{\mu Q_{\downarrow}(\mu)\mathcal{F}[g](\mu)}{R_{\downarrow}(\mu)\cosh\left[\frac{\pi\mu}{2}\right]} e^{-i\mu\bar{a}_N} \\ &= \int_{\mathbb{R}-i\kappa_\eta} \frac{d\mu}{4i^2\pi N \tau_N} \frac{Q_{\uparrow}(\mu)\mathcal{F}[g](\mu)}{R_{\uparrow}(\mu)\cosh\left[\frac{\pi\mu}{2}\right]} e^{-i\mu\bar{b}_N} - \int_{\mathbb{R}+i\kappa_\eta} \frac{d\mu}{4i^2\pi N \tau_N} \frac{\mu Q_{\downarrow}(\mu)\mathcal{F}[g](\mu)}{R_{\downarrow}(\mu)\cosh\left[\frac{\pi\mu}{2}\right]} e^{-i\mu\bar{a}_N} \\ &\quad - \frac{\mathcal{F}[g](i)}{\pi N \tau_N R_{\uparrow}(-i)} \{Q_{\uparrow}(-i)e^{-\bar{b}_N} - Q_{\downarrow}(i)e^{\bar{a}_N}\} \mathbf{1}_{1<\zeta}. \end{aligned} \quad (2.90) $$

In the second line we have deformed the integration contour to $\mathbb{R} - i\kappa_\eta$ for the first integral and to $\mathbb{R} + i\kappa_\eta$. Note that, in the process, one only picks poles of $1/\cosh[\pi\mu/2]$ at $\pm i$, and this provided that $1 < \zeta$. This generates the contribution of the last line, which also uses the inversion relations for $R_{\uparrow/\downarrow}$ and the parity of $\mathcal{F}[g]$.
Owing to the $O(\mu^{-k})$ for any $k \ge 0$ decay of the integrand at infinity, one readily then estimates the first integral to be $O(e^{-\kappa_{\eta}\bar{b}_{N}/N\tau_{N}})$ and the second one to be $O(e^{\kappa_{\eta}\bar{a}_{N}/N\tau_{N}})$. Finally, one has $Q_{\uparrow/\downarrow}(\mu) = 1 + O(e^{-\zeta(1-\eta)\bar{x}_{N}})$ uniformly in $\mu \in \mathbb{R} \pm i\kappa_{\eta}$ and for $\mu = \pm i$. Thus, all-in-all,

$$ \mathcal{J}[w'_{N}] = - \frac{\mathcal{F}[g](i)}{\pi N \tau_N R_{\uparrow}(-i)} \{e^{-\bar{b}_N} - e^{-\bar{a}_N}\} \mathbf{1}_{1<\zeta} + \frac{1}{N \tau_N} \mathcal{O}([e^{-\bar{b}_N} + e^{-\bar{a}_N}]e^{-\zeta(1-\eta)\bar{x}_N} + e^{-\kappa_{\eta}\bar{b}_N} + e^{\kappa_{\eta}\bar{a}_N}) . \quad (2.91) $$

It is clear from the previous handlings and the properties of $Q_{\uparrow/\downarrow}$ that the remainder is smooth in $a_N, b_N$ and that each $a_N$ or $b_N$ derivative of the remainder worsens the control by a factor of $\tau_N$.

The result then follows upon putting together all the estimates and exact expressions.

**Proposition 2.6.** It holds

$$ \int_{a_N}^{b_N} d\xi \, \mathcal{W}_N[V'_{N;\alpha}](\xi) = \frac{-\alpha}{2\pi N} \chi_{11;-(0)\chi_{12;-(0)}} - \frac{\alpha}{4i\pi N} \left\{ i(e^{\bar{b}_N} - e^{-\bar{a}_N}) \left( \frac{\chi_{11;-(0)\chi_{11}(i)}}{2} - \chi_{12;-(0)\chi_{11}(i)} \right) \right. \\ \left. + (e^{\bar{b}_N} + e^{-\bar{a}_N}) [\chi_{11;-(0)\chi_{12}(i)} - \chi_{12;-(0)\chi_{11}(i)} - \frac{i}{2}\chi_{11;-(0)\chi_{11}(i)}] \right\} \\ + \frac{\mathcal{F}[g](i)}{2\pi^2 N R_{\downarrow}(i) R_{\downarrow}(0)} (e^{-\bar{b}_N} + e^{-\bar{a}_N}) \mathbf{1}_{1<\zeta} + \mathcal{O}\left( \frac{e^{\kappa_{\eta}\bar{a}_N} + e^{-\kappa_{\eta}\bar{b}_N}}{N} \right) . \quad (2.92) $$

$\kappa_{\eta}$ appearing above is as introduced in (2.71).

*Proof—*

One starts from the partially integrated expression for $\mathcal{W}_N[V'_{N;\alpha}]$ obtained in Proposition 2.4. Then, with the notation of that proposition, one has

$$ \int_{a_N}^{b_N} d\xi \, \mathcal{W}_N[V'_{N;\alpha}](\xi) = \sum_{a=1}^{3} \int_{a_N}^{b_N} d\xi \, \varpi_N^{(a)}(\xi) . \quad (2.93) $$

Since,

$$ \int_{a_N}^{b_N} d\xi e^{-i\lambda\tau_N(\xi-a_N)} = \frac{1 - e^{-i\lambda\bar{x}_N}}{i\lambda\tau_N}, \quad (2.94) $$

one gets that

$$ \int_{a_N}^{b_N} d\xi \, \varpi_N^{(2)}(\xi) = - \frac{\alpha \chi_{12;-(0)}}{2\pi N} \int_{\mathbb{R}+2i\epsilon'} \frac{d\lambda}{2i\pi} \frac{\chi_{11}(\lambda) - \chi_{11;-(\lambda)}}{\lambda^2} . \quad (2.95) $$

There, we have used that $\chi_{1a}$ admits and analytic continuation from $\mathbb{H}^-$ to $\mathbb{H}^+$ denoted $\chi_{1a;-(\lambda)} = e^{-i\lambda\bar{x}_N}\chi_{1a}(\lambda)$ for $\mathfrak{J}(\lambda) > 0$. The integral is well defined in that the integrand behaves as $O(|\lambda|^{-5/2})$ at $\infty$. Splitting it in two pieces and taking the integral involving $\chi_{11}$ by means of the residues at the poles located above $\mathbb{R} + 2i\epsilon'$ -there are none- and the integral involving $\chi_{11; -}$ by means of the residues at the poles located below of $\mathbb{R} + 2i\epsilon'$ -there is only one at $\lambda = 0$- one gets that

$$ \int_{a_N}^{b_N} d\xi \, \varpi_N^{(2)}(\xi) = - \frac{\alpha \chi_{12;-(0)}}{2\pi N} \partial_{\lambda} \chi_{11;-(0)}. \quad (2.96) $$
We next focus on the contribution involving $\omega_N^{(1)}$. One gets

$$ \int_{a_N}^{b_N} d\xi \, \omega_N^{(1)}(\xi) = -\frac{r}{4\pi N} \int_{\mathbb{R}+2i\epsilon'} \frac{d\lambda}{2i\pi} \left\{ \frac{\chi_{12}(\lambda) - \chi_{12};-(\lambda)}{\lambda^2} \mathcal{U}_{12}(\lambda) + \frac{\chi_{11}(\lambda) - \chi_{11};-(\lambda)}{\lambda} \mathcal{U}_{11}(\lambda) \right\}. \quad (2.97) $$

Splitting it in two pieces and taking the integral involving $\chi_{11}$ by means of the residues at the poles located above $\mathbb{R}+2i\epsilon'$ - there are none- and the integral involving $\chi_{11};-$ by means of the residues at the poles located below of $\mathbb{R}+2i\epsilon'$ - there is only one at $\lambda=0$ - one gets that

$$ \int_{a_N}^{b_N} d\xi \, \omega_N^{(1)}(\xi) = -\frac{r}{4\pi N} \left\{ \partial_\lambda(\chi_{12};-\mathcal{U}_{12})(0) + \chi_{11};-(0)\mathcal{U}_{11}(0) \right\}. \quad (2.98) $$

A direct calculation leads to

$$ \mathcal{U}_{11}(0) = -iNu_N\chi_{12}(i) - \chi_{11}(i)\frac{Nu_N - v_N}{2}, \quad \mathcal{U}_{12}(0) = -v_N\chi_{11}(i) \quad \text{and} \quad \mathcal{U}_{12}'(0) = iNu_N\chi_{11}(i). \quad (2.99) $$

Here, we remind that $u_N$ and $v_N$ have been introduced in (2.64). Upon inserting the above into the closed expression for the integral of $\omega_N^{(1)}$, one eventually gets

$$ \int_{a_N}^{b_N} d\xi \, \omega_N^{(1)}(\xi) = \frac{r}{4\pi N} \left\{ iNu_N \left[ \chi_{11};-(0)\chi_{12}(i) - \chi_{12};-(0)\chi_{11}(i) - \frac{i}{2}\chi_{11};-(0)\chi_{11}(i) \right] \right. \\ \left. - v_N \left[ \frac{1}{2}\chi_{11};-(0)\chi_{11}(i) - \chi_{12};-(0)\chi_{11}(i) \right] \right\}. \quad (2.100) $$

Finally, we focus on estimating the contribution issuing from $\omega_N^{(3)}$. Starting from the expression for the integrand provided in Proposition 2.4 along with (2.94), one gets that

$$ \int_{a_N}^{b_N} d\xi \, \omega_N^{(3)}(\xi) = I_{\downarrow}^{(3)} + I_{\uparrow}^{(3)} + I_{0}^{(3)}. \quad (2.101) $$

There, one has

$$ \begin{aligned} I_{\downarrow}^{(3)} &= \frac{1}{4\pi N} \int_{\mathbb{R}+2i\epsilon'} \frac{d\lambda}{2i\pi\lambda^2} \left( [\mathbf{E}_L^{(\uparrow)}(\lambda) + (1-e^{i\lambda\bar{x}_N})\mathbf{E}_L^{(\downarrow)}(\lambda)] - e^{-i\lambda\bar{x}_N}\mathbf{E}_L^{(\uparrow)}(\lambda), \mathcal{E}^{(\downarrow)}(\lambda) \right) \\ &= \frac{1}{4\pi N} \int_{\mathbb{R}+i\kappa\eta'} \frac{d\lambda}{2i\pi\lambda^2} \left( \mathbf{E}_L^{(\uparrow)}(\lambda) + (1-e^{i\lambda\bar{x}_N})\mathbf{E}_L^{(\downarrow)}(\lambda), \mathcal{E}^{(\downarrow)}(\lambda) \right) + \mathbf{1}_{1<\zeta} \frac{e^{\bar{x}_N \mathcal{F}[g](i)}}{2\pi^2 N} (\mathbf{E}_L^{(\uparrow)}(i), \mathbf{E}_R^{(\downarrow)}(i)) \\ &\quad + \frac{1}{4\pi N} \partial_\lambda \left\{ e^{-i\lambda\bar{x}_N} (\mathbf{E}_L^{(\uparrow)}(\lambda), \mathcal{E}^{(\downarrow)}(\lambda)) \right\}_{|\lambda=0} - \frac{1}{4\pi N} \int_{\mathbb{R}-i\kappa\eta'} \frac{d\lambda}{2i\pi\lambda^2} e^{-i\lambda\bar{x}_N} (\mathbf{E}_L^{(\uparrow)}(\lambda), \mathcal{E}^{(\downarrow)}(\lambda)). \end{aligned} \quad (2.102) $$

Above, $\eta' > \eta$, and is taken small enough. Also, we have made use of the relation $(\mathbf{E}_L^{(\downarrow)}(i), \mathbf{E}_R^{(\downarrow)}(i)) = 0$.
To estimate the various contributions more precisely, one needs the auxiliary estimates

$$ (E_L^{(\uparrow)}(\lambda), E_R^{(\downarrow)}(\mu)) = \frac{-\mu}{R_\uparrow(\lambda)R_\downarrow(\mu)} + O\left(\sqrt{(1+|\lambda|)(1+|\mu|)} \cdot e^{-\zeta(1-\eta)\bar{x}_N}\right) \quad (2.103) $$

and

$$ (E_L^{(\downarrow)}(\lambda), E_R^{(\uparrow)}(\mu)) = O\left(\sqrt{(1+|\lambda|)(1+|\mu|)} \cdot e^{-\zeta(1-\eta)\bar{x}_N}\right). \quad (2.104) $$

Note that the remainders are holomorphic in $\lambda$ located between $\Gamma_\uparrow$ and $\Gamma_\downarrow$ and smooth in $a_N, b_N$, with each derivative adding a $\tau_N$ factor to the control. From that and the fact that $\mathcal{F}[g]$ is a Schwartz function, one infers the bounds

$$ (E_L^{(\uparrow)}(\lambda), \mathcal{E}^{(\downarrow)}(\lambda)) = -e^{\bar{a}_N} \frac{2i\mathcal{F}[g](i)}{\pi(i-\lambda)R_\uparrow(\lambda)R_\downarrow(i)} \mathbf{1}_{1<\zeta} + \frac{1}{R_\uparrow(\lambda)} O\left(\frac{e^{\bar{a}_N \chi_\eta}}{\sqrt{1+|\lambda|}}\right) + O\left(\frac{e^{\bar{a}_N \chi_\eta} e^{-\zeta(1-\eta)\bar{x}_N}}{\sqrt{1+|\lambda|}}\right) \quad (2.105) $$

so that the first remainder has a zero at $\lambda = 0$, and and

$$ (E_L^{(\downarrow)}(\lambda), \mathcal{E}^{(\uparrow)}(\lambda)) = O\left(\frac{e^{\bar{a}_N \chi_\eta}}{\sqrt{1+|\lambda|}}\right). \quad (2.106) $$

There, the remainders enjoy the same properties as above. Inserting these bounds inside of the obtained representation for $I_\downarrow^{(3)}$, one gets

$$ I_\downarrow^{(3)} = e^{\bar{a}_N} \frac{\mathcal{F}[g](i)}{2\pi^2 N R_\downarrow(i)} \mathbf{1}_{1<\zeta} \left\{ \frac{1}{i R_\uparrow(i)} + \frac{1}{R_\downarrow(0)} \right\} + O\left(\frac{e^{\bar{a}_N \chi_\eta}}{N}\right). \quad (2.107) $$

The remainder is now only $C^1$ in respect to $a_N, b_N$ and partial $a_N$ or $b_N$ derivatives thereof enjoy the same control with a $\tau_N$ additional factor.

Similarly, one obtains

$$ \begin{aligned} I_\uparrow^{(3)} &= \frac{-1}{4\pi N} \int_{\mathbb{R}+2i\epsilon'} \frac{d\lambda}{2i\pi\lambda^2} \left( [E_L^{(\uparrow)}(\lambda) + (1-e^{i\lambda\bar{x}_N})E_L^{(\downarrow)}(\lambda)] - e^{-i\lambda\bar{x}_N}E_L^{(\uparrow)}(\lambda), \mathcal{E}^{(\uparrow)}(\lambda) \right) \\ &= \frac{-1}{4\pi N} \int_{\mathbb{R}+i\chi_\eta'} \frac{d\lambda}{2i\pi\lambda^2} \left( [E_L^{(\uparrow)}(\lambda) + (1-e^{i\lambda\bar{x}_N})E_L^{(\downarrow)}(\lambda), \mathcal{E}^{(\uparrow)}(\lambda)] - \mathbf{1}_{1<\zeta} \frac{e^{-\bar{b}_N-\bar{x}_N}\mathcal{F}[g](i)}{2\pi^2 N} (E_L^{(\uparrow)}(-i), E_R^{(\downarrow)}(-i)) \right. \\ & \qquad \left. - \frac{1}{4\pi N} \partial_\lambda \left\{ e^{-i\lambda\bar{x}_N} (E_L^{(\uparrow)}(\lambda), \mathcal{E}^{(\uparrow)}(\lambda)) \right\}_{|\lambda|=0} + \frac{1}{4\pi N} \int_{\mathbb{R}-i\chi_\eta'} \frac{d\lambda}{2i\pi\lambda^2} e^{-i\lambda\bar{x}_N} (E_L^{(\uparrow)}(\lambda), \mathcal{E}^{(\uparrow)}(\lambda)) \right\}. \end{aligned} \quad (2.108) $$

As before, one gets the auxiliary estimates

$$ (E_L^{(\uparrow)}(\lambda), E_R^{(\downarrow)}(\mu)) = O\left(\sqrt{(1+|\lambda|)(1+|\mu|)} \cdot e^{-\zeta(1-\eta)\bar{x}_N}\right) \quad (2.109) $$

and

$$ (E_L^{(\downarrow)}(\lambda), E_R^{(\uparrow)}(\mu)) = \frac{\lambda}{R_\downarrow(\lambda)R_\uparrow(\mu)} + O\left(\sqrt{(1+|\lambda|)(1+|\mu|)} \cdot e^{-\zeta(1-\eta)\bar{x}_N}\right). \quad (2.110) $$
Again, the remainders are holomorphic in $\lambda$ located between $\Gamma_\uparrow$ and $\Gamma_\downarrow$ and smooth in $a_N, b_N$, with each derivative adding a $\tau_N$ factor to the control. Thus,

$$ (E_L^{(\uparrow)}(\lambda), \mathcal{E}^{(\uparrow)}(\lambda)) = O\left(\frac{e^{-\bar{b}_N - \zeta(1-\eta)\bar{x}_N}}{\sqrt{1+|\lambda|}}\right) \qquad (2.111) $$

and

$$ (E_L^{(\downarrow)}(\lambda), \mathcal{E}^{(\uparrow)}(\lambda)) = \frac{2\lambda \mathcal{F}[g](i) e^{-\bar{b}_N}}{\pi(i+\lambda)R_{\downarrow}(\lambda)R_{\uparrow}(-i)} \mathbf{1}_{1<\zeta} + O\left(\frac{e^{-\bar{b}_N \kappa_\eta}}{\sqrt{1+|\lambda|}}\right). \qquad (2.112) $$

There, the remainders enjoy the same properties as above. Hence, one gets that

$$ I_{\uparrow}^{(3)} = - \frac{\mathcal{F}[g](i) e^{-\bar{b}_N}}{2\pi^2 N R_{\uparrow}(-i)} \mathbf{1}_{1<\zeta} \int_{\mathbb{R}+i\kappa_{\eta}} \frac{d\lambda}{2i\pi\lambda R_{\downarrow}(\lambda)(i+\lambda)} + O\left(\frac{e^{-\kappa_{\eta}\bar{b}_N}}{N}\right). \qquad (2.113) $$

The remainder is $C^1$ in respect to $a_N, b_N$ and partial $a_N$ or $b_N$ derivatives thereof enjoy the same control with a $\tau_N$ additional factor. The remaining integral can be computed by means of taking the residues at $\lambda = 0$ and $\lambda = -i$ located below of $\mathbb{R} + i\kappa_\eta$, leading eventually to

$$ I_{\uparrow}^{(3)} = e^{-\bar{b}_N} \frac{\mathcal{F}[g](i)}{2\pi^2 N R_{\downarrow}(i)} \mathbf{1}_{1<\zeta} \left\{ \frac{1}{iR_{\uparrow}(i)} + \frac{1}{R_{\downarrow}(0)} \right\} + O\left(\frac{e^{-\kappa_{\eta}\bar{b}_N}}{N}\right). \qquad (2.114) $$

It remains to focus on $I_0^{(3)}$ which takes the form

$$ \begin{aligned} I_0^{(3)} &= \frac{1}{4\pi N} \int_{\mathbb{R}+2i\epsilon'} \frac{d\lambda}{2i\pi\lambda} \left( e^{-i\lambda\bar{a}_N} [E_L^{(\uparrow)}(\lambda) + (1-e^{i\lambda\bar{x}_N})E_L^{(\downarrow)}(\lambda)] - e^{-i\lambda\bar{b}_N} E_L^{(\uparrow)}(\lambda), E_R^{(\downarrow)}(\lambda) \right) \frac{\mathcal{F}[g](\lambda)}{\cosh\left[\frac{\pi\lambda}{2}\right]} \\ &= \frac{1}{4\pi N} \int_{\mathbb{R}+i\kappa_\eta} \frac{d\lambda}{2i\pi\lambda} e^{-i\lambda\bar{a}_N} (E_L^{(\uparrow)}(\lambda), E_R^{(\downarrow)}(\lambda)) \frac{\mathcal{F}[g](\lambda)}{\cosh\left[\frac{\pi\lambda}{2}\right]} - e^{\bar{a}_N} \frac{\mathcal{F}[g](i)}{2\pi^2 N} (E_L^{(\uparrow)}(i), E_R^{(\downarrow)}(i)) \mathbf{1}_{1<\zeta} \\ &\quad + \frac{\mathcal{F}[g](0)}{4\pi N} (E_L^{(\uparrow)}(0), E_R^{(\downarrow)}(0)) - e^{-\bar{b}_N} \frac{\mathcal{F}[g](i)}{2\pi^2 N} (E_L^{(\uparrow)}(-i), E_R^{(\downarrow)}(-i)) \mathbf{1}_{1<\zeta} \\ &\quad - \frac{1}{4\pi N} \int_{\mathbb{R}-i\kappa_\eta} \frac{d\lambda}{2i\pi\lambda} e^{-i\lambda\bar{b}_N} (E_L^{(\uparrow)}(\lambda), E_R^{(\downarrow)}(\lambda)) \frac{\mathcal{F}[g](\lambda)}{\cosh\left[\frac{\pi\lambda}{2}\right]}. \end{aligned} \qquad (2.115) $$

It is then enough to invoke the previous auxiliary bounds to infer that $(E_L^{(\uparrow)}(0), E_R^{(\downarrow)}(0)) = O(\mathcal{L}_N)$ while the two integral terms are a $N^{-1}O(e^{-\kappa_\eta \bar{b}_N} + e^{\kappa_\eta \bar{a}_N})$. Those auxiliary bounds also allow one to simplify the explicit contributions so that, up to subdominant corrections,

$$ I_0^{(3)} = \frac{i \mathcal{F}[g](i)}{2\pi^2 N R_{\uparrow}(i) R_{\downarrow}(i)} \mathbf{1}_{1<\zeta} (e^{-\bar{b}_N} + e^{\bar{a}_N}) + O\left(\frac{e^{-\kappa_\eta \bar{b}_N} + e^{\kappa_\eta \bar{a}_N}}{N}\right). \qquad (2.116) $$

The remainder is $C^1$ in respect to $a_N, b_N$ and partial $a_N$ or $b_N$ derivatives thereof enjoy the same control with a $\tau_N$ additional factor. By putting the three estimates together, we get

$$ \int_{a_N}^{b_N} d\xi \, \sigma_N^{(3)}(\xi) = \frac{\mathcal{F}[g](i) \mathbf{1}_{1<\zeta}}{2\pi^2 N R_{\downarrow}(0) R_{\uparrow}(i)} (e^{-\bar{b}_N} + e^{\bar{a}_N}) + O\left(\frac{e^{-\kappa_\eta \bar{b}_N} + e^{\kappa_\eta \bar{a}_N}}{N}\right). \qquad (2.117) $$
This entails the claim.

Below, we establish the unique solvability of the constraints (2.84) on the endpoints $a_N, b_N$ under the hypothesis that $b_N - a_N \ge \eta$, for some fixed $\eta > 0$. By the previous discussion, this implies that these unique solutions do correspond to the endpoints of the support of the equilibrium measure.

**Proposition 2.7.** Consider the subset of $\mathbb{R}^2$

$$ \mathcal{D}_{\zeta} = [\zeta; +\infty[\times] -\infty; -\zeta], \qquad (2.118) $$

with $\zeta > 0$ and small. For any $\zeta > 0$ there exists $N_0$ such that, for any $N \ge N_0$, there exists a unique solution $(b_{N;\alpha}, a_{N;\alpha}) \in \mathcal{D}_{\zeta}$ to the constraint equations (2.84). Moreover, it holds that

$$ b_{N;\alpha} = 1 + o(1) \quad \text{and} \quad a_{N;\alpha} = -1 + o(1). \qquad (2.119) $$

*Proof—*

It follows from Propositions 2.5-2.6 and the expansion $\chi_{11}(i) = -i/R_{\uparrow}(i) + O(e^{-\bar{x}_N \bar{x}_{\eta}})$ with a differentiable remainder, that the constraints (2.84) are equivalent to the system of equations for $a_N, b_N$:

$$ e^{\bar{b}_N} - e^{-\bar{a}_N} = 2 \frac{\alpha \chi_{11};-(0)}{r \chi_{11}(i)} + O\left(e^{-\bar{b}_N \bar{x}_{\eta}} + e^{\bar{a}_N \bar{x}_{\eta}}\right), \qquad (2.120) $$

and

$$ e^{\bar{b}_N} + e^{-\bar{a}_N} = \frac{4\pi N}{i r} \cdot \frac{1 + \frac{\alpha}{2\pi N} [\chi'_{11};-(0)\chi_{12};-(0) - \chi'_{12};-(0)\chi_{11};-(0) + \frac{1}{2}\chi_{11};-(0)]}{\chi_{11};-(0)\chi_{12}(i) - \chi_{12};-(0)\chi_{11}(i) - \frac{i}{2}\chi_{11};-(0)\chi_{11}(i)} + O\left(e^{\tilde{x}_{\eta}\bar{a}_N} + e^{-\tilde{x}_{\eta}\bar{b}_N}\right). \quad (2.121) $$

There, we agree upon $\tilde{x}_{\eta} = \min\{1, (1 - \eta)\zeta\}$.

A direct calculation based on the expansions (2.39) and (2.44) leads to

$$ \chi_{11};-(0)\chi_{12}(i) - \chi_{12};-(0)\chi_{11}(i) - \frac{i}{2}\chi_{11};-(0)\chi_{11}(i) = \frac{-1}{R_{\downarrow}(0)R_{\uparrow}(i)}\left(1 + ie^{-\bar{x}_N}\frac{R_{\uparrow}(i)}{R_{\downarrow}(i)} + O\left(e^{-\zeta(1-\eta)\bar{x}_N}\right)\right) \quad (2.122) $$

and

$$ \chi'_{11};-(0)\chi_{12};-(0) - \chi'_{12};-(0)\chi_{11};-(0) + \frac{1}{2}\chi_{11};-(0) = O(\bar{x}_N e^{-\zeta(1-\eta)\bar{x}_N}). \qquad (2.123) $$

This allows one to recast the second constraint in the form

$$ e^{\bar{b}_N} + e^{-\bar{a}_N} = Nc_0 \frac{1 + \frac{\bar{x}_N}{N} O\left(e^{\tilde{x}_{\eta}\bar{a}_N} + e^{-\tilde{x}_{\eta}\bar{b}_N} + e^{-\zeta(1-\eta)\bar{x}_N}\right)}{1 + ie^{-\bar{x}_N} \frac{R_{\uparrow}(i)}{R_{\downarrow}(i)} + O\left(e^{-\zeta(1-\eta)\bar{x}_N}\right)} = Nc_0 \cdot \left(1 + O\left(e^{\tilde{x}_{\eta}\bar{a}_N} + e^{-\tilde{x}_{\eta}\bar{b}_N}\right)\right). \quad (2.124) $$

There, we have set

$$ c_0 = \frac{4\pi}{r} \sqrt{\omega_1 + \omega_2 R_{\uparrow}(i)}. \qquad (2.125) $$
Finally, owing to

$$2 \frac{\alpha \chi_{11}(-0)}{\tau \chi_{11}(i)} = \frac{\alpha c_0}{\pi(\omega_1 + \omega_2)} \frac{1 + O(e^{-\zeta(1-\eta)\bar{x}_N})}{1 - ie^{-\bar{x}_N} \frac{R_\uparrow(i)}{R_\downarrow(i)} + O(e^{-\zeta(1-\eta)\bar{x}_N})} \quad (2.126)$$

one recasts the first constrain in the form

$$\begin{aligned} e^{\bar{b}_N} - e^{-\bar{a}_N} &= \frac{\alpha c_0}{\pi(\omega_1 + \omega_2)} \frac{1 + O(e^{-\zeta(1-\eta)\bar{x}_N})}{1 - ie^{-\bar{x}_N} \frac{R_\uparrow(i)}{R_\downarrow(i)}} + O(e^{-\bar{b}_N \tilde{z}_\eta} + e^{-\bar{a}_N \tilde{z}_\eta}) \\ &= \frac{\alpha c_0}{\pi(\omega_1 + \omega_2)} + O(e^{\tilde{z}_\eta \bar{a}_N} + e^{-\tilde{z}_\eta \bar{b}_N}). \end{aligned} \quad (2.127)$$

In order to prove more efficiently the existence and uniqueness for $N$ large enough of the system's solutions on the domain $\mathcal{D}_\zeta$ introduced in (2.118), it is convenient to pass to the finite in $N$-variables $(u_N, v_N)$ defined through (2.64).

Note that upon defining

$$\Psi(x, y) = \left( \frac{e^{\tau_N x} + e^{-\tau_N y}}{N}, e^{\tau_N x} - e^{-\tau_N y} \right), \quad (2.128)$$

one has that

$$\tilde{\mathcal{D}}_\zeta = \Psi(\mathcal{D}_\zeta) = \{ (u, v) \in \mathbb{R}^+ \times \mathbb{R} : u \ge \frac{1}{N^{1-\zeta}} \text{ and } Nu - 2N^\zeta \ge |v| \}. \quad (2.129)$$

Then, one may recast the constraints in the form

$$u_N = c_0 + \delta\Phi_1(u_N, v_N) \quad \text{and} \quad v_N = \frac{\alpha c_0}{\pi(\omega_1 + \omega_2)} + \delta\Phi_2(u_N, v_N). \quad (2.130)$$

The functions $\delta\Phi_a$ are $C^1$ on $\tilde{\mathcal{D}}_\zeta$ since the remainders in (2.124) and (2.127) are $C^1$ on $\mathcal{D}_\zeta$. Moreover, it is direct to estimate that throughout $\tilde{\mathcal{D}}_\zeta$, it holds

$$\delta\Phi_a(u, v) = O\left(\left(\frac{2}{Nu+v}\right)^{\tilde{z}_\eta} + \left(\frac{2}{Nu-v}\right)^{\tilde{z}_\eta}\right) = O(N^{-\zeta\tilde{z}_\eta}). \quad (2.131)$$

Taken that the remainder's estimates also hold for the first derivatives up to additional $\tau_N$ factors, one gets that

$$|\partial_u \delta\Phi_a(u, v)| + |\partial_v \delta\Phi_a(u, v)| = O(N^{-\zeta\tilde{z}_\eta(1-\eta')}), \quad (2.132)$$

with $\eta' > 0$ and small enough. Thus, introducing the $C^1$ diffeomorphism on $\tilde{\mathcal{D}}_\zeta$

$$\Phi(u, v) = \left( \frac{c_0 - u + \delta\Phi_1(u, v)}{\frac{\alpha c_0}{\pi(\omega_1 + \omega_2)} - v + \delta\Phi_2(u, v)} \right) \quad (2.133)$$

one has, uniformly throughout $\tilde{\mathcal{D}}_\zeta$, that $D\Phi = -I_2 + O(N^{-\zeta\tilde{z}_\eta(1-\eta')})$. One is thus in the setting where one can invoke the local inversion theorem so as to ensure that $\Phi$, as soon as $N$ is large enough, is a local $C^1$ diffeomorphism on $\tilde{\mathcal{D}}_\zeta$ such that, for any $(u, v) \in \tilde{\mathcal{D}}_\zeta$, there exist $N$-independent $s, s' > 0$ such that

$$\Phi : B_{(u,v),s} \to \Phi(B_{(u,v),s}) \supset B_{\Phi(u,v),s'} \quad (2.134)$$
is a diffeomorphism, with $B_{a,r}$ being the open ball of radius $r$ centred at $a$.

This is enough so as to ensure that $\Phi$ is a $C^1$ diffeomorphism on $\tilde{\mathcal{D}}_\zeta$. Indeed, assume that there exist $(u, v), (u', v') \in \tilde{\mathcal{D}}_\zeta, (u, v) \neq (u', v')$ such that $\Phi(u, v) = \Phi(u', v')$. Then, one has the relation

$$ \left\{ \begin{array}{l} u' - u = \delta\Phi_1(u, v) - \delta\Phi_1(u', v') = O(N^{-\zeta\bar{x}_\eta(1-\eta')}) \\ v' - v = \delta\Phi_2(u, v) - \delta\Phi_2(u', v') = O(N^{-\zeta\bar{x}_\eta(1-\eta')}) \end{array} \right. . \qquad (2.135) $$

However, $\Phi$ is injective on $B_{(u,v),s} \ni (u', v')$ what entails $(u, v) = (u', v')$, a contradiction. This entail that $\Phi$ is a diffeomorphism on $\tilde{\mathcal{D}}_\zeta$.

Finally, by the estimates on $\delta\Phi$, it holds that $\Phi(\zeta_0, \frac{\alpha\zeta_0}{\pi(\omega_1+\omega_2)}) = O(N^{-\zeta\bar{x}_\eta(1-\eta')})$. However, since there exists $s, s' > 0$ such that

$$ \Phi(B(\zeta_0, \frac{\alpha\zeta_0}{\pi(\omega_1+\omega_2)}, s), B(\Phi(\zeta_0, \frac{\alpha\zeta_0}{\pi(\omega_1+\omega_2)}, s'), \qquad (2.136) $$

it follows that $(0, 0) \in \Phi(\tilde{\mathcal{D}}_\zeta)$, what ensures the existence and uniqueness of solutions to the system (2.84) on $\mathcal{D}_\zeta$. The form of the leading large-$N$ behaviour for $a_{N;\alpha}, b_{N;\alpha}$ then follows readily. ■

**Lemma 2.8.** The following large-$N$ asymptotics hold

$$ e^{\bar{b}_{N;\alpha}} = \frac{\zeta_0 N}{2} + \frac{\alpha \zeta_0}{2\pi(\omega_1 + \omega_2)} - \frac{2i R_{\uparrow}(i)}{N \zeta_0 R_{\downarrow}(i)} \left(1 + \frac{2\mathcal{F}[g](i)}{\pi r} \mathbf{1}_{1<\zeta}\right) + O\left(\frac{1}{N^{\kappa_{\eta}}}\right) \qquad (2.137) $$

and

$$ e^{-\bar{a}_{N;\alpha}} = \frac{\zeta_0 N}{2} - \frac{\alpha \zeta_0}{2\pi(\omega_1 + \omega_2)} - \frac{2i R_{\uparrow}(i)}{N \zeta_0 R_{\downarrow}(i)} \left(1 + \frac{2\mathcal{F}[g](i)}{\pi r} \mathbf{1}_{1<\zeta}\right) + O\left(\frac{1}{N^{\kappa_{\eta}}}\right) \qquad (2.138) $$

These asymptotic expansions involve the constant

$$ \zeta_0 = \frac{4\pi}{r} \sqrt{\omega_1 + \omega_2} R_{\uparrow}(i) \qquad (2.139) $$

while $\kappa_\eta$ is as introduced in (2.71). Moreover, it also holds that

$$ \bar{b}_{N;\alpha} = \ln\left(\frac{\zeta_0 N}{2}\right) + \frac{\alpha}{\pi N(\omega_1 + \omega_2)} - \frac{1}{N^2} \left\{ \frac{4i R_{\uparrow}(i)}{\zeta_0^2 R_{\downarrow}(i)} \left(1 + \frac{2\mathcal{F}[g](i)}{\pi r} \mathbf{1}_{1<\zeta}\right) + \frac{\alpha^2}{2\pi^2(\omega_1 + \omega_2)^2} \right\} + O\left(\frac{1}{N^{\kappa_{\eta}+1}}\right) \qquad (2.140) $$

and

$$ \bar{a}_{N;\alpha} = -\ln\left(\frac{\zeta_0 N}{2}\right) + \frac{\alpha}{\pi N(\omega_1 + \omega_2)} + \frac{1}{N^2} \left\{ \frac{4i R_{\uparrow}(i)}{\zeta_0^2 R_{\downarrow}(i)} \left(1 + \frac{2\mathcal{F}[g](i)}{\pi r} \mathbf{1}_{1<\zeta}\right) - \frac{\alpha^2}{2\pi^2(\omega_1 + \omega_2)^2} \right\} + O\left(\frac{1}{N^{\kappa_{\eta}+1}}\right) \qquad (2.141) $$

**Proof—**

Starting from the expressions of Propositions 2.5 and 2.6, one infers that

$$ e^{\bar{b}_N} - e^{-\bar{a}_N} = 2 \frac{\alpha \chi_{11;-(0)}}{r \chi_{11}(i)} \cdot \left\{ 1 + \frac{2i \mathcal{F}[g](i) e^{-\bar{x}_N}}{\pi r \chi_{11}(i) R_{\uparrow}(-i)} \mathbf{1}_{1<\zeta} \right\}^{-1} + O\left(e^{-\bar{b}_N \kappa_{\eta}} + e^{\bar{a}_N \kappa_{\eta}}\right), \qquad (2.142) $$
and

$$
\begin{equation}
\begin{split}
e^{\bar{b}_N} + e^{-\bar{a}_N} = \frac{4\pi N}{i r} \cdot \frac{1 + \frac{\alpha}{2\pi N} [\chi'_{11};-(0)\chi_{12};-(0) - \chi'_{12};-(0)\chi_{11};-(0) + \frac{1}{2}\chi_{11}^2;-(0) \cdot \left\{1 + \frac{2i\mathcal{F}[g](i)e^{-\bar{x}_N}}{\pi r \chi_{11}(i)R_\uparrow(-i)} \mathbf{1}_{1<\zeta}\right\}^{-1}]}{\chi_{11};-(0)\chi_{12}(i) - \chi_{12};-(0)\chi_{11}(i) - \frac{i}{2}\chi_{11};-(0)\chi_{11}(i) - \frac{2i\mathcal{F}[g](i)e^{-\bar{x}_N}}{\pi r R_\uparrow(i)R_\downarrow(0)} \mathbf{1}_{1<\zeta}} \\
\qquad + O\left(e^{-\bar{b}_N \chi_\eta} + e^{\bar{a}_N \chi_\eta}\right).
\end{split}
\tag{2.143}
\end{equation}
$$

Then, by using the asymptotic expansions given in (2.45) and (2.40) one gets that

$$
\begin{equation}
\begin{split}
e^{\bar{b}_N} + e^{-\bar{a}_N} = N c_0 \frac{1 - \frac{\alpha}{\pi N} \cdot \frac{i R_{\uparrow}(i)}{R_{\downarrow}(i)} \cdot \frac{2 \mathcal{F}[g](i)}{\pi \tau(\omega_1 + \omega_2)} e^{-\bar{x}_N} \mathbf{1}_{1<\zeta} + \frac{1}{N} \mathcal{O}\left(\alpha e^{-\zeta(1-\eta)\bar{x}_N} + \alpha e^{-\bar{x}_N} \mathbf{1}_{1<\zeta}\right)}{1 + \frac{i R_{\uparrow}(i)}{R_{\downarrow}(i)} e^{-\bar{x}_N} \left[1 + \frac{2 \mathcal{F}[g](i)}{\pi \tau} \mathbf{1}_{1<\zeta}\right] + \mathcal{O}\left(\alpha e^{-\zeta(1-\eta)\bar{x}_N}\right)} \\
\qquad + \mathcal{O}\left(e^{-\bar{b}_N \zeta_\eta} + e^{\bar{a}_N \zeta_\eta}\right),
\end{split}
\tag{2.144}
\end{equation}
$$

and

$$
e^{\bar{b}_N} - e^{-\bar{a}_N} = \frac{\alpha c_0}{\pi(\omega_1 + \omega_2)} + O\left(e^{-\bar{b}_N \zeta_\eta} + e^{\bar{a}_N \zeta_\eta}\right). \quad (2.145)
$$

From there on, the result follows from straightforward algebra.

<figure><img src="image_3.png" /></figure>

3 Large- *N* behaviour of the interpolating integral

**Proposition 3.1. It holds**

$$
\int_{a_{N;\alpha}}^{b_{N;\alpha}} d\xi \, \xi \, \widehat{\varrho}_{eq;\alpha}(\xi) = \frac{\alpha}{\pi(\omega_1 + \omega_2)N\tau_N} \left[ \ln\left(\frac{Nc_0}{2}\right) + i \ln'R_{\downarrow}(0) \right] + O\left(\frac{\tau_N}{N} \left[ N^{-1} \mathbf{1}_{1<\zeta} + N^{-\kappa_\eta} \right] \right) \quad (3.1)
$$

Proof —

One starts by observing that

$$
\int_{a_{N;\alpha}}^{b_{N;\alpha}} d\xi \, e^{-i\lambda\tau_N(\xi - a_{N;\alpha})} \xi = \frac{a_{N;\alpha} - b_{N;\alpha}e^{-i\lambda\bar{x}_{N;\alpha}}}{i\lambda\tau_N} + \frac{e^{-i\lambda\bar{x}_{N;\alpha}} - 1}{(\lambda\tau_N)^2}. \quad (3.2)
$$

Here, we agree that $\bar{x}_{N;\alpha} = \bar{b}_{N;\alpha} - \bar{a}_{N;\alpha}$. Thus, according to the partially integrated expression for $W_N[V'_{N;\alpha}]$ obtained in Proposition 2.4, one may decompose the integral into three terms

$$
\int_{a_{N;\alpha}}^{b_{N;\alpha}} d\xi \, \xi \, \widehat{\varrho}_{eq;\alpha}(\xi) = \sum_{a=1}^{3} \mathcal{H}^{(a)} \qquad (3.3)
$$

where

$$
\mathcal{H}^{(1)} = \frac{r\tau_N}{4i\pi N} \int_{\mathbb{R}+2ie'} \frac{d\lambda}{2i\pi} \left\{ \frac{\chi_{12}(\lambda)}{\lambda} \mathcal{W}_{12}(\lambda) + \chi_{11}(\lambda) \mathcal{W}_{11}(\lambda) \right\} \cdot \left\{ \frac{a_{N;\alpha} - b_{N;\alpha}e^{-i\lambda\bar{x}_{N;\alpha}}}{i\lambda\tau_N} + \frac{e^{-i\lambda\bar{x}_{N;\alpha}} - 1}{(\lambda\tau_N)^2} \right\}, \quad (3.4)
$$
$$ \mathcal{H}^{(2)} = \frac{\alpha \tau_N}{2i\pi N} \int_{\mathbb{R}+2i\epsilon'} \frac{d\lambda}{2i\pi} \chi_{11}(\lambda)\chi_{12;-(0)} \cdot \left\{ \frac{a_{N;\alpha} - b_{N;\alpha}e^{-i\lambda\bar{x}_{N;\alpha}}}{i\lambda^2\tau_N} + \frac{e^{-i\lambda\bar{x}_{N;\alpha}} - 1}{\lambda^3\tau_N^2} \right\}, \quad (3.5) $$

and, by employing the notations introduced in Proposition 2.4

$$ \mathcal{H}^{(3)} = \mathcal{H}_{\uparrow}^{(3)} + \mathcal{H}_{\downarrow}^{(3)} + \mathcal{H}_{0}^{(3)} \quad \text{with} \quad \mathcal{H}_{\upsilon}^{(3)} = \int_{a_{N;\alpha}}^{b_{N;\alpha}} d\xi \, \xi \, \omega_{N;\upsilon}^{(3)}(\xi), \quad \upsilon \in \{\uparrow, \downarrow, 0\}. \quad (3.6) $$

We stress that now, all the above quantities involve the solution $\chi$ subordinate to the choice of endpoints $a_{N;\alpha}$, $b_{N;\alpha}$.

The first two contributions can be computed in closed form. Indeed, by applying the previously introduced notations, one gets

$$ \mathcal{H}^{(2)} = \frac{\alpha \tau_N}{2i\pi N} \chi_{12;-(0)} \int_{\mathbb{R}+2i\epsilon'} \frac{d\lambda}{2i\pi} \cdot \left\{ \frac{a_{N;\alpha} \chi_{11}(\lambda) - b_{N;\alpha} \chi_{11;-(\lambda)}}{i\lambda^2 \tau_N} + \frac{\chi_{11;-(\lambda)} - \chi_{11}(\lambda)}{\lambda^3 \tau_N^2} \right\}. \quad (3.7) $$

Due to the $O(\lambda^{-5/2})$ decay of the integrand at $\infty$

i) the contribution of the integrand involving $\chi_{11}$ can be evaluated by taking the residues of the integrand's poles located above $\mathbb{R}+2i\epsilon'$. Since there are no poles, this part produces 0.

ii) The contribution of the integrand involving $\chi_{11;-(\lambda)}$ can be evaluated by taking the residues of the integrand's poles located below $\mathbb{R}+2i\epsilon'$. The only poles present are the third and second order poles at 0.

All-in-all, one gets that

$$ \mathcal{H}^{(2)} = -\frac{\alpha \tau_N}{2i\pi N} \chi_{12;-(0)} \cdot \left\{ -\frac{b_{N;\alpha}}{i\tau_N} \chi'_{11;-(0)} + \frac{1}{2\tau_N^2} \chi''_{11;-(0)} \right\}. \quad (3.8) $$

The large-$N$ behaviour of $\chi_{1a}$ given in (2.44), allows one to infer that

$$ \chi_{12;-(0)} = O(e^{-\zeta(1-\eta)\bar{x}_{N;\alpha}}) \quad \text{and} \quad \chi_{11;-(0)}^{(k)} = O(\bar{x}_{N;\alpha}^k) \quad \text{with} \quad k \in \mathbb{N}. \quad (3.9) $$

This entails that

$$ \mathcal{H}^{(2)} = O\left(\frac{|\alpha|\tau_N}{N} \cdot e^{-\zeta(1-\eta)\bar{x}_{N;\alpha}}\right) = O\left(\frac{|\alpha|\tau_N}{N^{1+2\zeta(1-\eta)}}\right). \quad (3.10) $$

Similar handlings lead to

$$ \mathcal{H}^{(1)} = -\frac{\tau_N}{4i\pi N} \left\{ -\frac{b_{N;\alpha}}{i\tau_N} \partial_\lambda(\chi_{12;-(\mathcal{U}_{12})}(0)) + \frac{1}{2\tau_N^2} \partial_\lambda^2(\chi_{12;-(\mathcal{U}_{12})}(0)) \right. \\ \left. - \frac{b_{N;\alpha}}{i\tau_N} (\chi_{11;-(\mathcal{U}_{11})}(0)) + \frac{1}{\tau_N^2} \partial_\lambda(\chi_{11;-(\mathcal{U}_{11})}(0)) \right\}. \quad (3.11) $$
A long but straightforward calculation utilising the expansion (2.44) yields

$$
\mathcal{H}^{(1)} = \frac{r\tau_N}{4i\pi N R_{\downarrow}(0) R_{\uparrow}(i)} \left\{ \begin{aligned}
& \bar{b}_{N;\alpha} e^{\bar{b}_{N;\alpha}} + \bar{a}_{N;\alpha} e^{-\bar{a}_{N;\alpha}} - (e^{\bar{b}_{N;\alpha}} - e^{-\bar{a}_{N;\alpha}}) \cdot (1 - i \ln'R_{\downarrow}(0)) \\
& + i \frac{R_{\uparrow}(i)}{R_{\downarrow}(i)} \cdot \left[ \bar{b}_{N;\alpha} e^{-\bar{b}_{N;\alpha}} + \bar{a}_{N;\alpha} e^{\bar{a}_{N;\alpha}} - e^{-\bar{x}_{N;\alpha}}(e^{\bar{b}_{N;\alpha}} - e^{-\bar{a}_{N;\alpha}}) \cdot (1 + i \ln'R_{\downarrow}(0)) \right]
\end{aligned} \right\} \left\{ (1 + O(\tau_N e^{-\zeta(1-\eta)\bar{x}_{N;\alpha}})) \right\}. \quad (3.12)
$$

Observe that, for large N, one has

$$
\bar{b}_{N;\alpha} e^{\bar{b}_{N;\alpha}} + \bar{a}_{N;\alpha} e^{-\bar{a}_{N;\alpha}} = v_{N;\alpha} \ln \left( \frac{Nu_{N;\alpha}e}{2} \right) + O(N^{-2}) \quad (3.13)
$$

and

$$
\bar{b}_{N;\alpha} e^{-\bar{b}_{N;\alpha}} + \bar{a}_{N;\alpha} e^{\bar{a}_{N;\alpha}} = O\left(\frac{v_{N;\alpha} \tau_N}{(Nu_{N;\alpha})^2}\right). \quad (3.14)
$$

Finally, by inserting the large-$N$ expansion of the endpoints obtained in Lemma 2.8, one gets

$$
\mathcal{H}^{(1)} = \frac{\alpha}{\pi(\omega_1 + \omega_2)N\tau_N} \left[ \ln\left(\frac{N\epsilon_0}{2}\right) + i \ln'R_{\downarrow}(0) \right] - \frac{2\alpha^2}{\pi^2(\omega_1 + \omega_2)^2 N^2 \tau_N} + O\left(\frac{1}{N^3} + \frac{e^{-\zeta(1-\eta)\bar{x}_{N;\alpha}}}{N}\right). \quad (3.15)
$$

Hence, to conclude, it remains to estimate $\mathcal{H}^{(3)}$. We estimate separately each of the $\mathcal{H}_v^{(3)}$. For $v \in \{\uparrow, \downarrow\}$, by taking the expressions for $\varpi_{N;\uparrow/\downarrow}^{(3)}$ obtained in Proposition 2.4 one gets the representation

$$
\mathcal{H}_v^{(3)} = \frac{\epsilon_v}{4i\pi N \tau_N} \int_{\mathbb{R}+2i\epsilon'} \frac{d\lambda}{2i\pi\lambda^3} (\mathbf{F}_{\uparrow}(\lambda) + e^{-i\lambda\bar{x}_N} \mathbf{F}_{\downarrow}(\lambda), \mathcal{E}^{(v)}(\lambda)), \quad (3.16)
$$

where we have introduced

$$
\mathbf{F}_{\downarrow}(\lambda) = (i\bar{b}_{N;\alpha}\lambda + 1)\mathbf{E}_{L}^{(\uparrow)}(\lambda) \quad (3.17)
$$

$$
\mathbf{F}_{\uparrow}(\lambda) = -(i\bar{a}_{N;\alpha}\lambda + 1)\mathbf{E}_{L}^{(\uparrow)}(\lambda) - (i\bar{b}_{N;\alpha}\lambda + 1)\mathbf{E}_{L}^{(\downarrow)}(\lambda) + e^{i\lambda\bar{x}_{N;\alpha}}(i\bar{a}_{N;\alpha}\lambda + 1)\mathbf{E}_{L}^{(\uparrow)}(\lambda). \quad (3.18)
$$

Thus, upon introducing $\tilde{\kappa}_{\eta} = (1 - \eta)\min\{1, \zeta\}$ and deforming the integration contours, one gets

$$
\mathcal{H}_v^{(3)} = \frac{\epsilon_v}{4i\pi N \tau_N} \int_{\mathbb{R}+i\tilde{\kappa}_\eta} \frac{d\lambda}{2i\pi\lambda^3} (\mathbf{F}_\uparrow(\lambda), \mathcal{E}^{(v)}(\lambda)) - \frac{\epsilon_v}{8i\pi N \tau_N} \partial_\lambda^2 \left\{ e^{-i\lambda\bar{x}_N} (\mathbf{F}_\downarrow(\lambda), \mathcal{E}^{(v)}(\lambda)) \right\}_{|\lambda=0} \\
+ \frac{\epsilon_v}{4i\pi N \tau_N} \int_{\mathbb{R}-i\tilde{\kappa}_\eta} \frac{d\lambda}{2i\pi\lambda^3} e^{-i\lambda\bar{x}_N} (\mathbf{F}_\downarrow(\lambda), \mathcal{E}^{(v)}(\lambda)) \quad (3.19)
$$

Then, using that within the band $|\mathfrak{J}(\lambda)| \le \tilde{\kappa}_\eta$ one has the bounds

$$
\| \mathcal{E}^{(v)}(\lambda) \| \le \frac{C}{1 + |\lambda|} (N^{-1} \mathbf{1}_{1<\zeta} + N^{-\kappa_\eta}), \quad (3.20)
$$

one readily gets that

$$
|\mathcal{H}_v^{(3)}| \le \frac{C \tau_N}{N} (N^{-1} \mathbf{1}_{1<\zeta} + N^{-\kappa_\eta}). \qquad (3.21)
$$
Finally, we focus on $\mathcal{H}_0^{(3)}$ which, upon using that $(E_L^{(\downarrow)}(\lambda), E_R^{(\downarrow)}(\lambda)) = 0$, may be recast as

$$ \mathcal{H}_0^{(3)} = \frac{1}{4i\pi N \tau_N} \int_{\mathbb{R}+2i\epsilon'} \frac{d\lambda}{2i\pi\lambda^2} \frac{\mathcal{F}[g](\lambda)}{\cosh[\frac{\pi\lambda}{2}]} (E_L^{(\uparrow)}(\lambda), E_R^{(\downarrow)}(\lambda)) \{(i\bar{a}_{N;\alpha}\lambda + 1)e^{-i\lambda\bar{a}_{N;\alpha}} - e^{-i\lambda\bar{b}_{N;\alpha}}(i\bar{b}_{N;\alpha}\lambda + 1)\}. \quad (3.22) $$

Then, it is enough to observe that throughout the strip $|\mathfrak{J}(\lambda)| \le \tilde{\kappa}_\eta$

$$ (E_L^{(\uparrow)}(\lambda), E_R^{(\downarrow)}(\lambda)) = -\frac{\lambda}{R(\lambda)} + \mathfrak{R}_N(\lambda) \quad \text{with} \quad \mathfrak{R}_N(\lambda) = O((1+|\lambda|)e^{-\zeta(1-\eta)\bar{x}_{N;\alpha}}). \quad (3.23) $$

In particular, the leading term has a second order zero at $\lambda = 0$. This leads to

$$ \mathcal{H}_0^{(3)} = \int_{\mathbb{R}+i\tilde{\kappa}_\eta} \frac{d\lambda}{2i\pi\lambda^2} \frac{\mathcal{F}[g](\lambda)}{\cosh[\frac{\pi\lambda}{2}]} (E_L^{(\uparrow)}(\lambda), E_R^{(\downarrow)}(\lambda)) \frac{i\bar{a}_{N;\alpha}\lambda + 1}{4i\pi N \tau_N} e^{-i\lambda\bar{a}_{N;\alpha}} \\ - \int_{\mathbb{R}-i\tilde{\kappa}_\eta} \frac{d\lambda}{2i\pi\lambda} \frac{\mathcal{F}[g](\lambda)}{\cosh[\frac{\pi\lambda}{2}]R(\lambda)} \frac{i\bar{b}_{N;\alpha}\lambda + 1}{4i\pi N \tau_N} e^{-i\lambda\bar{b}_{N;\alpha}} + \int_{\mathbb{R}+2i\epsilon'} \frac{d\lambda}{2i\pi\lambda^2} \frac{\mathcal{F}[g](\lambda)}{\cosh[\frac{\pi\lambda}{2}]} \mathfrak{R}_N(\lambda) \frac{i\bar{b}_{N;\alpha}\lambda + 1}{4i\pi N \tau_N} e^{-i\lambda\bar{b}_{N;\alpha}}. \quad (3.24) $$

Upon deforming further the integrals up to $\mathbb{R} \pm i\tilde{\kappa}_\eta$ and picking the residues of the simple pole at $\pm i$ and then applying direct bounds, one eventually gets

$$ |\mathcal{H}_0^{(3)}| \le \frac{C}{N\tau_N} \{\tau_N[N^{-1}\mathbf{1}_{1<\zeta} + N^{-\tilde{\kappa}_\eta}] + \tau_N e^{-\zeta(1-\eta)\bar{x}_{N;\alpha} + 2\epsilon'\bar{b}_{N;\alpha}}\} \le \frac{C}{N}[N^{-1}\mathbf{1}_{1<\zeta} + N^{-\tilde{\kappa}_\eta}]. \quad (3.25) $$

The claim then follows by putting the various estimates together. ■

## Conclusion

In this work, we have provided a full characterisation of the equilibrium measure which governs the leading asymptotic expansion of the logarithm of the Lukyanov integral. This allowed us to check, by means of explicit calculations the predictions relative to the leading term of the Lukyanov conjecture describing the large-$N$ behaviour of a multiple-integral supposed to provide the lattice regularisation of the vacuum expectation value of the exponential of the field operator in the quantum Sinh-Gordon finite volume $R$ field theory. Our calculations confirm this part of the conjecture. However, a lack of sharp bounds on the remainder, issuing from our incapacity to control, on sufficiently fine scales, the inverse of the master operator arising in the system of loop equations, does not allow us to prove that indeed the other corrections which could contribute to the asymptotics of the derivative $\partial_\alpha \ln \mathcal{Z}_N[V_{N;\alpha}]$ will not do so on a stronger than $\ln N$ scale. It would be extremely interesting to develop a much better understanding of the scaling regimes of the master operator appropriate for this setting.

We plan to address these questions, in full rigour, by alternative methods in further works.

## Acknowledgements

KKK is supported by the CNRS. CDG and KKK are supported by the ERC Project LDRAM: ERC-2019-ADG Project 884584 and by the CNRS 80Prime Grant "Asymptotiques d'intégrales multiples associées à la séparation des variables quantiques". The authors thank Alice Guionnet for numerous stimulating discussions related to the topics tackled in this paper.
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