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# Landscape k-complexity of isotropic centered Gaussian fields

Jean-Marc Azaïs, Céline Delmas

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Jean-Marc Azaïs, Céline Delmas. Landscape k-complexity of isotropic centered Gaussian fields. 2025. hal-05005220

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Preprint submitted on 25 Mar 2025

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# Landscape k-complexity of isotropic centered Gaussian fields

Jean-Marc Azaïs¹ and Céline Delmas²

¹IMT, Université de Toulouse, France., e-mail: jean-marc.azais@math.univ-toulouse.fr

²MIAT, INRAE, Université de Toulouse, France., e-mail: celine.delmas.toulouse@inrae.fr

**Abstract:** In large dimension, we study the asymptotic behavior of the mean number of critical points with index $k$ below a level $u$ for an isotropic centered Gaussian random field defined on a family of subsets of $\mathbb{R}^d$ depending on $d$. We prove the existence of three regimes depending on the speed of growth of the volume the parameter set. In the first regime the mean number of critical points decreases exponentially with the dimension. For the second regime, there exists a critical level $u_c$ such that the mean number of critical points with index $k$ below a level $u$ with $u > u_c$ increases exponentially with the dimension $d$ independently of the index $k$ and decreases exponentially with $d$ when $u < u_c$. In the third regime, there exists a layered structure depending on the level $u$ considered and on the index $k$ of the critical points. This behavior is similar to the one encountered on the sphere by Auffinger et al. [5]. In the particular case of the Bargmann-Fock field, only two regimes coexist.

**MSC2020 subject classifications:** Primary 60G15, 60G10.

**Keywords and phrases:** Complexity, Critical points, Gaussian fields, High dimension.

## 1. Introduction

This note studies the asymptotic behavior in large dimension of the mean number of critical points with index $k$ below a level $u$. This for centered Gaussian isotropic random fields defined on a family of subsets $T_d$ of $\mathbb{R}^d$ whose volume $|T_d|$ is logarithmically comparable to that of a ball as $d$ tends to infinity. More precisely, we assume that

$$ \frac{1}{d} \log \left( \frac{|T_d|}{\kappa_d} \right) \to V \in \mathbb{R} \text{ as } d \to +\infty, $$

where $\kappa_d$ denotes the volume of the unit ball in $\mathbb{R}^d$. $T_d$ may be, for example, the ball of radius $R$ in $\mathbb{R}^d$ or the square $[0, L/\sqrt{d}]^d$ with $0 < L, R < +\infty$. In those cases $R := \exp(V)$ and $L := \sqrt{2\pi} \exp(V + 1/2)$. We prove that there exists a critical value $V = V_c^1$, or equivalently a critical radius $R_c^1$ for the ball of radius $R$ or a critical side length $L_c^1$ for the square $[0, L/\sqrt{d}]^d$, such that the mean number of critical points with index $k$ on $T_d$ decreases exponentially with $d$ when $V < V_c^1$ ($R < R_c^1$ or $L < L_c^1$), is logarithmically comparable to zero when $V = V_c^1$ ($R = R_c^1$ or $L = L_c^1$) and increases exponentially with $d$ when $V > V_c^1$ ($R > R_c^1$ or $L > L_c^1$). We obtain

$$ V_c^1 = \log \left( \frac{\exp(1)\sqrt{3\lambda_2}}{\sqrt{\lambda_4}} \right), \quad R_c^1 = \frac{\exp(1)\sqrt{3\lambda_2}}{\sqrt{\lambda_4}} \quad \text{and} \quad L_c^1 = \exp(3/2)\sqrt{\frac{6\lambda_2\pi}{\lambda_4}}, $$

where $\lambda_2$ and $\lambda_4$ are respectively the second and fourth spectral moments of the Gaussian field. When $V > V_c^1$, there also exist another critical value $V_c^2 > V_c^1$ such that when $V_c^1 < V < V_c^2$ there exist a critical level $v_c$ for which the mean number of critical points with index $k$ below the level $u = v_c\sqrt{d}$ on $T_d$, $E(Crt_{u\downarrow}^k(T_d))$, decreases exponentially with $d$ when $v < v_c$ and increases exponentially with $d$ when $v > v_c$. This behaviour is independent on the index $k$. When $V > V_c^2$ there exist critical levels $v_c^0 < v_c^1 < v_c^2 < \dots$ for which the mean number of critical points with index $k$ below the level $u = v_c\sqrt{d}$
on $T_d$ decreases exponentially with $d$ when $v < v_c^k$ and increases exponentially with $d$ when $v > v_c^k$. Thus there exists a layered structure in the distribution of the critical points: there will be approximatively no critical points below $u_0 = v_c^0 \sqrt{d}$, only critical points with index 0 between $u_0$ and $u_1 = v_c^1 \sqrt{d}$, only critical points with indexes 0 and 1 between $u_1$ and $u_2 = v_c^2 \sqrt{d}$ and so on. We have, as $d \to \infty$,

$$ \mathbb{E}(\mathrm{Crt}_{u\downarrow}^k(T_d)) \approx \exp(d \times \Theta_k(V, v)), $$

where $\approx$ mean equivalence of logarithms and $\Theta_k(V, v)$ is the landscape $k$-complexity of the Gaussian random field on $T_d$. The logarithmic behavior of $\mathbb{E}(\mathrm{Crt}_{u\downarrow}^k(T_d))$ as a function of $d$ is related to the sign of $\Theta_k(V, v)$.

Our study is analogous to that of [5] and [4] that considers the landscape $k$-complexity of centered isotropic Gaussian random fields defined on the high-dimensional sphere. The particularity of our study is that we consider Gaussian random fields defined on “flat” subsets $T_d$ of $\mathbb{R}^d$. We prove the existence of two critical values $V_c^1$ and $V_c^2$ for the volume of $T_d$ corresponding to three different regimes for the behavior of $\mathbb{E}(\mathrm{Crt}_{u\downarrow}^k(T_d))$. These critical values do not appear in the study of [5] and [4]. Our main tools are the Kac-Rice formula and the Large Deviations Principle for the $k$th smallest eigenvalue of a GOE-matrix, as in [5] and [4]. The classical papers devoted to the “flat case”, see [14] and [11], use a quadratic penalty. Our approach is in some sense similar to those approaches. For example, the quadratic penalty used in [14] can be interpreted as a constraint that limits the parameters to be in a sphere. But precise equations are different, as well as they are different from the “round case” of [5] and [4]. So, it is worth establishing these equations in this short note. In addition, papers such as [14] and [11] do not specify the index.

Our main result is Theorem 3.2. It describes the logarithmic behavior of the number of critical points as a function of the level. In contrast to [5] and [4], there is no distinction between “pure-like” and general models. The three behaviors can occur depending on the size of the set. The Bargmann-Fock field corresponds to the limit case $V_c^1 = V_c^2$. Therefore only two regimes coexist for the behavior of $\mathbb{E}(\mathrm{Crt}_{u\downarrow}^k(T_d))$ depending on the volume of $T_d$.

## 2. Assumptions, notations and background

In this paper $X(\cdot) = \{X(t) : t \in \mathbb{R}^d\}$ is a real-valued centered stationary and isotropic Gaussian field with unit variance. Its covariance function is defined by

$$ \mathbb{E}(X(s), X(t)) = c(s, t) = \mathbf{r}(\|s - t\|^2). \qquad (1) $$

$\mathbf{r}$ is assumed to be $\mathbb{C}^2$. The spectral moments $\lambda_0, \lambda_2$ and $\lambda_4$ are defined by

$$ 1 = \mathrm{Var}(X(t)) = \mathbf{r}(0), $$

$$ \lambda_{2n} := \mathrm{Var} \left( \frac{\partial^n X(t)}{\partial t^\ell} \right) = (-1)^n \frac{(2n)!}{n!} \mathbf{r}^{(n)}(0) \text{ with } \ell = 1, \dots, d \text{ and } n = 1, 2. \qquad (2) $$

The covariance function $\mathbf{r}(\cdot)$ is assumed to define a valid covariance in any dimension. It will be called a “Schoenberg covariance”. Proposition 2.1 below gives a characterization of such covariances.

**Proposition 2.1 ([15]).** A covariance function defined by $\mathbf{r}(1)$ is a Schoenberg covariance with $\mathbf{r}(0) = 1$ if and only if there exists a probability measure $G$ on $[0, +\infty)$ such that

$$ \mathbf{r}(x) = \int_{0}^{+\infty} e^{-xw} G(dw) \quad \text{for all } x \ge 0. \qquad (3) $$
A Schoenberg covariance satisfies $\lambda_4 \ge 3\lambda_2^2$ when these quantities are well defined (see [9]). Equality holds if and only if: $\mathbf{r}(t) = e^{-at}$ for some $a > 0$. In that case $\mathbb{E}(X(0)X(t)) = e^{-a\|t\|^2}$. This random field is named “Bargmann-Fock field”.

In this paper, we also assume that the paths of $X(\cdot)$ are $\mathbb{C}^2$ and $\lambda_2 > 0$. Then, the sample paths are almost surely Morse [3] and this implies that the number of critical points with index $k$ in a Borel set $T$, $\mathrm{Crt}^k(T)$, is well defined. We recall that the index is the number of negative eigenvalues of the Hessian denoted here by $i(X''(t))$. More precisely, we define

$$ \mathrm{Crt}^k(T) := \#\{t \in T : X'(t) = 0, i(X''(t)) = k\}. \qquad (4) $$

We define also the number of critical points of index $k$ below the level $u$ by

$$ \mathrm{Crt}_{u\downarrow}^k(T) := \#\{t \in T : X'(t) = 0, i(X''(t)) = k, X(t) < u\}. \qquad (5) $$

First we recall the following proposition, [7, Proposition 4], giving the exact expressions of the mean number of critical points with index $k$ below a level $u$ on a set $T$.

**Proposition 2.2.** Let $L_p$ be the $p$th ordered eigenvalue of a $(d+1)$-GOE matrix ($L_1 \le L_2 \le \dots \le L_{d+1}$). Under assumptions above,

$$ \mathbb{E} (\mathrm{Crt}^k(T)) = \frac{\sqrt{2}|T|}{\pi^{(d+1)/2}} \left(\frac{\lambda_4}{3\lambda_2}\right)^{d/2} \Gamma\left(\frac{d+1}{2}\right) \mathbb{E} \left(\exp\left(-\frac{L_{k+1}^2}{2}\right)\right). \qquad (6) $$

When $\lambda_4 - 3\lambda_2^2 > 0$,

$$ \mathbb{E} (\mathrm{Crt}_{u\downarrow}^k(T)) = \frac{\sqrt{2}|T|}{\pi^{(d+1)/2}} \left(\frac{\lambda_4}{3\lambda_2}\right)^{d/2} \Gamma\left(\frac{d+1}{2}\right) \times \mathbb{E} \left\{ \exp\left(-\frac{L_{k+1}^2}{2}\right) \Phi \left[ \sqrt{\frac{\lambda_4}{\lambda_4 - 3\lambda_2^2}} \left(u - L_{k+1} \frac{\sqrt{6}\lambda_2}{\sqrt{\lambda_4}}\right) \right] \right\}, \quad (7) $$

and when $\lambda_4 - 3\lambda_2^2 = 0$

$$ \mathbb{E} (\mathrm{Crt}_{u\downarrow}^k(T)) = \frac{\sqrt{2}|T|\lambda_2^{d/2}}{\pi^{(d+1)/2}} \Gamma\left(\frac{d+1}{2}\right) \mathbb{E} \left( \exp\left(-\frac{L_{k+1}^2}{2}\right) \mathbf{1}_{L_{k+1} \le u/\sqrt{2}} \right). \qquad (8) $$

**Remark.** This kind of result was first obtained in the spherical case by [4] and [5]. This result can also be found in [13, Proposition 3.9] written in a slightly different way.

### 3. Logarithmic asymptotics

In this section, we consider the logarithmic asymptotic behavior of $\mathbb{E}(\mathrm{Crt}_{u\downarrow}^k(T_d))$ as $d \to \infty$, where $T_d$ is a family of Borel sets. Roughly speaking, we obtain the same kind of results as in [4] and [5] when the volume of the sets $T_d$ is logarithmically comparable to that of a ball. More precisely we assume:

$$ \frac{1}{d} \log \left( \frac{|T_d|}{\kappa_d} \right) \to V \in \mathbb{R} \text{ when } d \to +\infty, \qquad (9) $$
where $\kappa_d = \frac{\pi^{d/2}}{\Gamma(\frac{d}{2} + 1)}$ denotes the volume of the unit ball in $\mathbb{R}^d$. $T_d$ may be, for example, the ball of radius $R$ in $\mathbb{R}^d$ or the square $[0, L/\sqrt{d}]^d$ with $0 < L < +\infty$. In these cases $R$ and $L$ are given by

$$R := \exp(V) \quad \text{and} \quad L := \exp(V + 1/2)\sqrt{2\pi} \qquad (10)$$

We set $\tilde{L}_{k+1} := \frac{L_{k+1}}{\sqrt{d+1}}$ for $k = 0, \dots, d$. The Large Deviations Principle (LDP) for the $(k+1)$ smallest eigenvalue of a $d$-GOE matrix obtained in [5, Theorem A.9], formulated below as a lemma, is the main tool for the results of this section.

**Lemma 3.1.** Let $L_1 \le L_2 \le \dots \le L_{d+1}$ be the ordered eigenvalues of a $(d+1)$-GOE matrix. We set $\tilde{L}_{k+1} := \frac{L_{k+1}}{\sqrt{d+1}}$, then $\tilde{L}_{k+1}$ satisfies the LDP with speed $(d+1)$ and good rate function

$$J_k(x) = (k+1)I_1(-x, 2^{-1/2}), \qquad (11)$$

with

$$I_1(x, \sigma) := \begin{cases} \int_{2\sigma}^{x} \sigma^{-1} \sqrt{\left(\frac{z}{2\sigma}\right)^2 - 1} dz, & \text{if } x \ge 2\sigma, \\ +\infty, & \text{otherwise.} \end{cases} \qquad (12)$$

### 3.1. Main result

We assume $\lambda_4 - 3\lambda_2^2 > 0$ with the condition (9) on the set $T_d$. We set

$$v := \frac{u}{\sqrt{d+1}}, \qquad E_{\star} := -\left(\frac{\lambda_4 + 3\lambda_2^2}{\lambda_2\sqrt{3\lambda_4}}\right), \qquad E_{\star\star} := -\frac{2\sqrt{3}\lambda_2}{\sqrt{\lambda_4}}. \qquad (13)$$

When $E_{\star}^2 - (k+1)^2(E_{\star\star} - E_{\star})^2 \ne 0$, we define

$$x_{k,v} := \frac{-\sqrt{2v}E_{\star} + \sqrt{2}(k+1)(E_{\star\star} - E_{\star})\sqrt{(k+1)^2(E_{\star\star} - E_{\star})^2 + (v^2 - E_{\star}^2)}}{E_{\star}^2 - (k+1)^2(E_{\star\star} - E_{\star})^2}, \qquad (14)$$

and when $E_{\star}^2 - (k+1)^2(E_{\star\star} - E_{\star})^2 = 0$,

$$x_{k,v} := -\frac{E_{\star}^2 + v^2}{\sqrt{2v}E_{\star}}. \qquad (15)$$

$$\Theta_k^*(V, v) := \begin{cases} V + \frac{1}{2} \log \left( \frac{\lambda_4}{3\lambda_2} \right) - 1 & \text{if } v \ge E_{\star\star} \\ V + \frac{1}{2} \log \left( \frac{\lambda_4}{3\lambda_2} \right) - \left( 1 - \frac{(v - E_{\star\star})^2}{E_{\star\star}(E_{\star\star} - E_{\star})} \right) & \text{if } E_{\star} \le v \le E_{\star\star} \\ V + \frac{1}{2} \log \left( \frac{\lambda_4}{3\lambda_2} \right) - \left( \frac{x_{k,v}^2}{2} - \frac{(v + x_{k,v} \frac{E_{\star\star}}{\sqrt{2}})^2}{E_{\star\star}(E_{\star\star} - E_{\star})} \right) - J_k(x_{k,v}) & \text{if } v \le E_{\star} \end{cases} \qquad (16)$$

where the function $J_k$ is defined in (11).
**Theorem 3.2.** Under assumptions above, we have

$$ \lim_{d \to +\infty} \frac{1}{d} \log \mathbb{E} \left( \mathrm{Crt}^k (T_d) \right) = \Theta^*(V) := V + \frac{1}{2} \log \left( \frac{\lambda_4}{3\lambda_2} \right) - 1 \quad \text{and} \\ \lim_{d \to +\infty} \frac{1}{d} \log \mathbb{E} \left( \mathrm{Crt}_{u\downarrow}^k (T_d) \right) = \Theta_k^*(V, v). $$

The proof of this theorem is given in Section 4.

**Remark.** We can define two critical values, $V_c^1$ and $V_c^2$ such that

$$ \Theta^*(V_c^1) = \Theta_k^*(V_c^1, E_{\star\star}) = 0 \Leftrightarrow V_c^1 := \log \left( \frac{\exp(1)\sqrt{3\lambda_2}}{\sqrt{\lambda_4}} \right) \qquad (17) $$

and

$$ \Theta_k^*(V_c^2, E_*) = 0 \Leftrightarrow V_c^2 := \log \left( \frac{\exp(E_*/E_{\star\star})\sqrt{3\lambda_2}}{\sqrt{\lambda_4}} \right). \qquad (18) $$

When the volume of $T_d$ is such that $V < V_c^1$ then $\Theta^*(V) < 0$; therefore the mean number of critical points with index $k$ on $T_d$ decreases exponentially with $d$. When $V > V_c^1$, this mean number increases exponentially with $d$.

**Lemma 3.3.** We have the following properties.

a. $E_\star < -2 < E_{\star\star} < 0$.

b. For $V$ and $k$ fixed, $\Theta_k^*(V, v)$ is a constant for $v$ above the level $E_{\star\star}$, a strictly increasing function of $v$ not depending on $k$ for $E_\star \le v < E_{\star\star}$ and a strictly increasing function of $v$ depending on $k$ below $E_\star$.

c. For $V$ and $v < E_\star$ fixed, $\Theta_k^*(V, v)$ is a strictly decreasing function of $k$.

The proof of Lemma 3.3 is given in Section 4.

**Remark.** 1. $\Theta_k^*(V, v)$ is a strictly increasing function of $V$ for $v$ and $k$ fixed. Therefore, because of (17) and (18), for $V_c^1 < V < V_c^2$, we have $\Theta_k^*(V, E_{\star\star}) > 0$ and $\Theta_k^*(V, E_\star) < 0$. Since $\Theta_k^*(V, v)$ is an strictly increasing function of $v$ for $E_\star < v < E_{\star\star}$ and $V$ and $k$ fixed, there exists a critical level $v_c$ such that $\Theta_k^*(V, v_c) = 0$. It is easy to check that

$$ v_c := E_{\star\star} - \sqrt{(V_c^1 - V) E_{\star\star} (E_{\star\star} - E_\star)}. \qquad (19) $$

When $v > v_c$, the mean number of critical points with index $k$ below the level $u = v\sqrt{d+1}$ increases exponentially with $d$. It decreases exponentially with $d$ when $v < v_c$ independently of $k$.

2. In the same way, we can check that for $V > V_c^2 > V_c^1$ we have $\Theta_k^*(V, E_\star) > 0$ and there exist critical values $v_c^0 < v_c^1 < v_c^2 < \dots < E_\star$ such that

$$ \Theta_k^*(V, v_c^k) = 0. $$

We also have

$$ \Theta_k^*(V, v) < 0 \text{ for } v < v_c^k \text{ and } \Theta_k^*(V, v) > 0 \text{ for } v > v_c^k. $$

Then the mean number of critical points with index $k$ below the level $u = v\sqrt{d+1}$ decreases exponentially with $d$ when $v < v_c^k$ and increases exponentially with $d$ when $v > v_c^k$. Therefore there exists a layered structure such that below the level $v_c^0$, there will be approximatively no critical points, between $v_c^0$ and $v_c^1$, only critical points with index 0, between $v_c^1$ and $v_c^2$, only critical points with indexes 0 and 1 and so on.
### 3.2. Example: the Matérn Gaussian field

In this section $X(\cdot)$ is a centered Matérn Gaussian random field with covariance function satisfying

$$ \mathbb{E}(X(s)X(t)) = c(s,t) = \mathbf{r}(\|s-t\|^2) := \frac{\sigma^2}{2^{\nu-1}\Gamma(\nu)} \left( \frac{\sqrt{2\nu}\|s-t\|^2}{\ell} \right)^{\nu} K_{\nu} \left( \frac{\sqrt{2\nu}\|s-t\|^2}{\ell} \right), $$

where $K_ν$ is the modified Bessel function of the second kind. This covariance function has three positive parameters $\sigma^2$, $\ell$ and $\nu$ controlling respectively the variance, the spatial range and the smoothness. We take $\sigma^2 := 1$. We assume $\nu > 2$. We have (see Proposition 2.1 in [12])

$$ \mathbf{r}'(0) = -\frac{\nu}{2(\nu-1)\ell^2} \quad \text{and} \quad \mathbf{r}''(0) = \frac{\nu^2}{4(\nu-1)(\nu-2)\ell^4}. $$

By (2), we deduce

$$ \lambda_2 = \frac{\nu}{(\nu - 1)\ell^2} \quad \text{and} \quad \lambda_4 = \frac{3\nu^2}{(\nu - 1)(\nu - 2)\ell^4}. $$

For $\nu > 2$ we have

$$ \lambda_4 - 3\lambda_2^2 = \frac{3\nu^2}{(\nu - 1)^2(\nu - 2)\ell^4} > 0. $$

By (17) and (18)

$$ V_c^1 = 1 - \frac{1}{2} \log \left( \frac{\nu}{\ell^2(\nu - 2)} \right) \quad \text{and} \quad V_c^2 = \frac{\nu - 3/2}{\nu - 2} - \frac{1}{2} \log \left( \frac{\nu}{\ell^2(\nu - 2)} \right). $$

By (10) we deduce the critical radii and lengths $R_c^1, R_c^2, L_c^1, L_c^2$. By (13) we have

$$ E_{\star} = - \frac{2\nu - 3}{(\nu - 1)^{1/2}(\nu - 2)^{1/2}} \quad \text{and} \quad E_{\star\star} = -2 \frac{(\nu - 2)^{1/2}}{(\nu - 1)^{1/2}}. $$

By (19) we obtain the critical level

$$ v_c = -2\sqrt{(\nu - 2)/(\nu - 1)} - \sqrt{2(V - V_c^1)/(\nu - 1)}. $$

Figure 1 displays the function $\Theta_k^*(V, v)$ when $\nu = 4$ and $\ell = 1$ for two different values of $V$. In that case $\lambda_2 = 4/3$, $\lambda_4 = 8$, $E_{\star} \simeq -2.04$, $E_{\star\star} \simeq -1.63$ and $V_c^1 \simeq 0.65$, $V_c^2 \simeq 0.90$. So we took $V = 0.75$ and $V = 3$ to illustrate the regimes $V_c^1 < V < V_c^2$ and $V > V_c^2$.

### 3.3. Particular case: the Bargmann-Fock field

In this section, we assume that $\lambda_4 - 3\lambda_2^2 = 0$. We have

$$ \mathbb{E}(X(0), X(t))) = \mathbf{r}(\|s-t\|^2) = \exp(-a\|s-t\|^2), \quad a > 0. \quad (20) $$

We define $E_{\infty} := -2$, and

$$ \tilde{\Theta}_k(V, v) := \begin{cases} V + \frac{1}{2} \log(\lambda_2) - 1 & \text{if } v \ge E_{\infty} \\ V + \frac{1}{2} \log(\lambda_2) - \frac{v^2}{4} - (k+1)I_1 \left( \frac{v\sqrt{2}}{E_{\infty}}, \frac{1}{\sqrt{2}} \right) & \text{if } v \le E_{\infty} \end{cases} $$

where $I_1$ is defined in (12).
<figure><img src="image_2.png" /><figcaption>FIG 1. Function $\Theta_k^*(V, v)$ for $V = 3/4$ (left) and $V = 3$ (right), $\mathbf{r}(\|s - t\|^2)$ being the Matérn covariance function with $v = 4$ and $\ell = 1$, $k = 0$ in black, $k = 1$ in green and $k = 100$ in red.</figcaption></figure>

**Theorem 3.4.** Under assumptions above and assuming that the family of sets $T_d$ satisfies (9), we have

$$ \lim_{d \to +\infty} \frac{1}{d} \log \mathbb{E} \left( \mathrm{Crt}^k(T_d) \right) = \tilde{\Theta}(V) := V + \frac{1}{2} \log(\lambda_2) - 1 $$

$$ \lim_{d \to +\infty} \frac{1}{d} \log \mathbb{E} \left( \mathrm{Crt}_{u\downarrow}^k(T_d) \right) = \tilde{\Theta}_k(V, v). $$

**Remark.** This theorem is the limit case of Theorem 3.2 for which $E_* = E_{**} = E_\infty = -2$. Its proof follows the same lines. There exists a critical value $V_c := \log(\frac{e}{\sqrt{\lambda_2}})$ such that for $V < V_c$ the mean number of critical points with index $k$ decreases exponentially with $d$ whereas, for $V > V_c$, it increases exponentially with $d$. In that case there also exist critical values $v_c^0 < v_c^1 < v_c^2 < \dots < -2$ such that $\tilde{\Theta}_k(V, v_c^k) = 0$ and we retrieve the layered structure as in the general case above.

Figure 2 displays the function $\tilde{\Theta}_k(V, v)$ when $a = 5$. In this case $V_c \simeq -0.15$ and we took $V = 0$ to illustrate the regime $V > V_c$.

## 4. Proofs

*Proof of Theorem 3.2.* By Proposition 2.2 we have

$$ \lim_{d \to +\infty} \frac{1}{(d+1)} \log \mathbb{E} \left( \mathrm{Crt}_{u\downarrow}^k (T_d) \right) = V + \frac{1}{2} \log \left( \frac{\lambda_4}{3\lambda_2} \right) + \lim_{d \to +\infty} \frac{1}{(d+1)} \log \mathbb{E} \left( e^{(d+1)g_d(\tilde{L}_{k+1})} \right) $$

where

$$ g_d(x) := -\frac{x^2}{2} + \frac{1}{(d+1)} \log \Phi \left( \sqrt{d+1} \sqrt{\frac{\lambda_4}{\lambda_4 - 3\lambda_2^2}} \left( v - x \frac{\sqrt{6\lambda_2}}{\sqrt{\lambda_4}} \right) \right). $$

It is easy to check that

$$ E_{**} - E_* = \frac{(\lambda_4 - 3\lambda_2^2)}{\lambda_2\sqrt{3\lambda_4}}. $$
<figure><img src="image_3.png" /><figcaption>FIG 2. Function $\tilde{\Theta}_k(V, v)$ for $\mathbf{r}(\|s-t\|^2) = \exp(-5\|s-t\|^2)$, $V=0$, $k=0$ in black, $k=1$ in green and $k=100$ in red.</figcaption></figure>

Then

$$g_d(x) = -\frac{x^2}{2} + \frac{1}{(d+1)} \log \Phi \left( \sqrt{\frac{2(d+1)}{E_{\star\star}(E_{\star}-E_{\star\star})}} \left(v + x \frac{E_{\star\star}}{\sqrt{2}}\right) \right).$$

When $v \ge E_{\star\star}$, for $x \le -\sqrt{2}$, we have $v + x \frac{E_{\star\star}}{\sqrt{2}} \ge 0$, then

$$\lim_{d \to +\infty} \frac{1}{(d+1)} \log \Phi \left( \sqrt{\frac{2(d+1)}{E_{\star\star}(E_{\star}-E_{\star\star})}} \left(v + x \frac{E_{\star\star}}{\sqrt{2}}\right) \right) = 0.$$

So

$$\lim_{d \to +\infty} \frac{1}{(d+1)} \log \mathbb{E} \left( e^{(d+1)g_d(\tilde{L}_{k+1})} \right) = \lim_{d \to +\infty} \frac{1}{(d+1)} \log \mathbb{E} \left( e^{(d+1)g(\tilde{L}_{k+1})} \right),$$

with

$$g(x) = -\frac{x^2}{2}.$$

Since $\tilde{L}_{k+1}$ satisfies the LDP with the good rate function $J_k(x) = (k+1)I_1(-x, 2^{-1/2})$, with $I_1$ defined in (12), using Laplace-Varadhan's lemma we obtain

$$\lim_{d \to +\infty} \frac{1}{(d+1)} \log \mathbb{E} \left( e^{(d+1)g(\tilde{L}_{k+1})} \right) = \sup_{x \le -\sqrt{2}} (g(x) - J_k(x)) = -1.$$

This gives the first line of $\Theta_k^*(V, v)$ in (16) and the expression of $\Theta^*(V)$ in Theorem 3.2.

When $v < E_{\star\star}$, for $x \le -\sqrt{2}$,

$$\lim_{d \to +\infty} \frac{1}{(d+1)} \log \Phi \left( \sqrt{\frac{2(d+1)}{E_{\star\star}(E_{\star}-E_{\star\star})}} \left(v + x \frac{E_{\star\star}}{\sqrt{2}}\right) \right) = \frac{1}{E_{\star\star}(E_{\star\star}-E_{\star})} \left(v + x \frac{E_{\star\star}}{\sqrt{2}}\right)^2 \mathbf{1}_{\{x \ge -\frac{v\sqrt{2}}{E_{\star\star}}\}}.$$
So

$$ \lim_{d \to +\infty} \frac{1}{(d+1)} \log \mathbb{E} \left( e^{(d+1)g_d(\tilde{L}_{k+1})} \right) = \lim_{d \to +\infty} \frac{1}{(d+1)} \log \mathbb{E} \left( e^{(d+1)\tilde{g}(\tilde{L}_{k+1})} \right), $$

with

$$ \tilde{g}(x) = -\frac{x^2}{2} + \frac{1}{E_{\star\star}(E_{\star\star} - E_{\star})} \left( v + x \frac{E_{\star\star}}{\sqrt{2}} \right)^2 \mathbf{1}_{\{x \geq -\frac{v\sqrt{2}}{E_{\star\star}}\}}. $$

Using Laplace-Varadhan's lemma we obtain

$$ \lim_{d \to +\infty} \frac{1}{(d+1)} \log \mathbb{E} \left( e^{(d+1)\tilde{g}(\tilde{L}_{k+1})} \right) = \sup_{x \leq -\sqrt{2}} (\tilde{g}(x) - J_k(x)). $$

It is easy to check that the maximum of $\tilde{g}(x) - J_k(x)$ is obtained on the interval $[-\frac{v\sqrt{2}}{E_{\star\star}}; -\sqrt{2}]$. In the case $E_{\star} < v < E_{\star\star}$, we can check that $\tilde{g}'(x) > 0$ and $-J'_{k}(x) > 0 \forall x \in [-\frac{v\sqrt{2}}{E_{\star\star}}; -\sqrt{2}]$. Thus the maximum of $\tilde{g}(x) - J_k(x)$ is obtained for $x = -\sqrt{2}$. This gives the second line of $\Theta_{\star}^{\star}(V, v)$ in (16).

When $v < E_{\star}$ we look for the maximum of $\tilde{g}(x) - J_k(x)$ on $[-\frac{v\sqrt{2}}{E_{\star\star}}; -\sqrt{2}]$. We have $\tilde{g}'(x) > 0$ and $-J'_{k}(x) > 0 \forall x \in [-\frac{v\sqrt{2}}{E_{\star\star}}; -\frac{v\sqrt{2}}{E_{\star}}]$. Moreover $\tilde{g}'(-\sqrt{2}) - J'_{k}(-\sqrt{2}) < 0$. So the maximum is reached on $[-\frac{v\sqrt{2}}{E_{\star}}; -\sqrt{2}]$

$$ \tilde{g}'(x) - J'_{k}(x) = 0 \Leftrightarrow (k+1)\sqrt{x^2-2} = -\frac{(xE_{\star} + \sqrt{2}v)}{(E_{\star\star} - E_{\star})}. \quad (21) $$

This implies that

$$ \left( \frac{E_{\star}^2}{(E_{\star\star} - E_{\star})^2} - (k+1)^2 \right) x^2 + \frac{2\sqrt{2}vE_{\star}}{(E_{\star\star} - E_{\star})^2} x + 2(k+1)^2 + \frac{2v^2}{(E_{\star\star} - E_{\star})^2} = 0. $$

When $E_{\star}^2 - (k+1)^2(E_{\star\star} - E_{\star})^2 \neq 0$, this equation admits two roots

$$ x_{\pm} = \frac{-\sqrt{2vE_{\star}} + \sqrt{2}(k+1)(E_{\star\star} - E_{\star})\sqrt{(k+1)^2(E_{\star\star} - E_{\star})^2 + (v^2 - E_{\star}^2)}}{E_{\star}^2 - (k+1)^2(E_{\star\star} - E_{\star})^2}. $$

When $E_{\star}^2 - (k+1)^2(E_{\star\star} - E_{\star})^2 < 0$, we have $x_{-} > 0$. The only suitable root is $x_{+}$. When $E_{\star}^2 - (k+1)^2(E_{\star\star} - E_{\star})^2 > 0$, we can check that $x_{-} < -\sqrt{2v}/E_{\star}$. The only suitable root is $x_{+}$.

When $E_{\star}^2 - (k+1)^2(E_{\star\star} - E_{\star})^2 = 0$, the root is

$$ x := -\frac{E_{\star}^2 + v^2}{\sqrt{2vE_{\star}}}. $$

That concludes the proof.

□

*Proof of Lemma 3.3.* Let us prove that $\Theta_{\star}^{\star}(V, \cdot)$ is strictly increasing on $(-\infty, E_{\star})$.

$$ \frac{\partial \Theta_{\star}^{\star}(V, v)}{\partial v} = -x_{k,v} \frac{\partial x_{k,v}}{\partial v} + \frac{(\sqrt{2}v + x_{k,v}E_{\star\star})(\sqrt{2} + E_{\star\star}\partial x_{k,v}/\partial v)}{E_{\star\star}(E_{\star\star} - E_{\star})} - J'_{k}(x_{k,v}) \frac{\partial x_{k,v}}{\partial v}. $$

By (21) we have

$$ -J'_{k}(x_{k,v}) = (k+1)\sqrt{x_{k,v}^2 - 2} = -\frac{(x_{k,v}E_{\star} + \sqrt{2}v)}{(E_{\star\star} - E_{\star})}. $$
We deduce, after calculations

$$ \frac{\partial \Theta_k^*(V, v)}{\partial v} = \sqrt{2} \frac{(\sqrt{2}v + x_{k,v}E_{**})}{E_{**}(E_{**} - E_{\star})}, $$

Since $x_{k,v} > -\frac{v\sqrt{2}}{E_{**}}$ (see the end of the proof of Theorem 3.2), we deduce the strict positivity of $\frac{\partial \Theta_k^*(V,v)}{\partial v}$ for $v \le E_{\star}$. The other elements of items a. and b. are easy to check.

Let us prove item c. Functions $J_k$, $x_{k,v}$ and $\Theta_k^*$ as functions of $k$ can be extended on real numbers. So we can derive with respect to $k$. By direct calculations we have

$$ \frac{\partial \Theta_k^*(V, v)}{\partial k} = \frac{\partial x_{k,v}}{\partial k} \left( \frac{x_{k,v}E_* + \sqrt{2}v + (k+1)(E_{**} - E_*)\sqrt{x_{k,v}^2 - 2}}{E_{**} - E_*} \right) - I_1(-x_{k,v}, \frac{1}{\sqrt{2}}). $$

By (21) we obtain

$$ \frac{\partial \Theta_k^*(V, v)}{\partial k} = -I_1(-x_{k,v}, \frac{1}{\sqrt{2}}) < 0. $$

□

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