On the expected number of critical points of
locally isotropic Gaussian random fields

Locally isotropic random fields on the $N$-dimensional Euclidean space $\mathbb{R}^{N}$ were introduced by Kolmogorov in 1941 [Ko41] for the application in statistical theory of turbulence. Since then, this class of stochastic processes has been extensively studied in both physics and mathematics. In particular, locally isotropic Gaussian random fields have been used to model a particle confined in a random potential and serve as a toy model for the elastic manifold. For an incomplete list of literature, we refer the interested reader to [MP91, En93, Fy04, FS07, FB08, FN12, FLD18, FLD20] for the background in physics and [Kli12, AZ20, AZ22, BBMsd, BBMsd2, XZ22] for the mathematical development.

The goal of this paper is two-fold. For application to statistical physics, frequently we need to send $N\to\infty$ for the thermodynamic limit, which puts additional restrictions on the class of such Gaussian fields. Using the Kac–Rice formula and the connection to random matrices, in the seminal work [Fy04] Fyodorov considered the expected number of critical points of isotropic Gaussian random fields on $\mathbb{R}^{N}$ in the asymptotic regime $N\to\infty$. This is commonly known as landscape complexity (or complexity for simplicity) in statistical physics; see also [FN12, Fy15] for related topics. Recently, Auffinger and Zeng provided a detailed study on the complexity of non-isotropic Gaussian random fields with isotropic increments [AZ20, AZ22]. On the other hand, for applications to statistics and other fields, it is also of interest to consider random fields on a finite-dimensional Euclidean space. Guided by this principle, Cheng and Schwartzman gave a representation of the expected number of critical points of isotropic Gaussian random fields on $\mathbb{R}^{N}$ with fixed $N$ [CS18, Theorem 3.5]. Here an apparent gap arises: What about the non-isotropic Gaussian fields on the fixed $\mathbb{R}^{N}$? We provide several answers to this question. Furthermore, we show that a technical assumption used in [AZ20] is redundant in the large $N$ limit as conjectured by the authors while it is useful to obtain a representation via matrices from the Gaussian Orthogonal Ensemble (GOE) in finite dimensions.

Let us be more precise. A locally isotropic Gaussian random field $H_{N}=\left\{H_{N}(x):x\in\mathbb{R}^{N}\right\}$ is a centered Gaussian process indexed by $\mathbb{R}^{N}$ that satisfies

Here the function $D_{N}\colon\mathbb{R}_{+}\to\mathbb{R}_{+}$ is called the structure function of $H_{N}$, $\|\cdot\|$ is the Euclidean norm and $\mathbb{R}_{+}=[0,\infty)$. The process $H_{N}$ is also known as a Gaussian random field with isotropic increments. The condition \reftagform@1.1 determines the law of $H_{N}$ up to an additive shift by a Gaussian random variable. Following [Ya87], we recall some basic properties of the structure function. Let $\mathcal{D}_{N}$ denote the set of all $N$-dimensional structure functions and $\mathcal{D}_{\infty}$ the set of structure functions which belong to $\mathcal{D}_{N}$ for all $N\in\mathbb{N}$. Since any $N$-dimensional structure function is necessarily an $M$-dimensional structure function for all integers $M$ less than $N$, it is clear that

where the symbol $\supset$ denotes inclusion. In the following, we write $D_{\infty}\in\mathcal{D}_{\infty}$ for the structure function of a field that can be defined on $\mathbb{R}^{N}$ for all natural numbers $N\in\mathbb{N}$ and in this case we frequently write $N=\infty$. Let us define

Here $J_{N}$ is the $N$th Bessel function of the first kind, which has the following representation

From here it can be shown that $\Lambda_{N}(\sqrt{2N}x)\stackrel{{\scriptstyle N\rightarrow\infty}}{{\longrightarrow}}e^{-x^{2}}$. By [Sch38, Ya57], locally isotropic Gaussian random fields (or equivalently the class $\mathcal{D}_{N}$) can be classified into two cases:

1. 1.

   Isotropic fields. There exists a function $B_{N}\colon\mathbb{R}_{+}\to\mathbb{R}$ such that

   $\displaystyle\mathbb{E}\left[H_{N}(x)H_{N}(y)\right]=B_{N}\left(\|x-y\|^{2}\right),$

   (1.2)

   where $B_{N}$ has the representation

   $\displaystyle B_{N}(r)=C_{N}+\int_{(0,\infty)}\Lambda_{N}(\sqrt{rt})\nu_{N}(\mathrm{d}t),$

   with a constant $C_{N}\in\mathbb{R}_{+}$ and a finite measure $\nu_{N}$ on $(0,\infty)$. Clearly, in this case we have $D_{N}(r)=2(B_{N}(0)-B_{N}(r))$. In particular, for the class $\mathcal{D}_{\infty}$, we write $B(r)=B_{\infty}(r)$ and it can be represented as

   $$B(r)=C+\int_{(0,\infty)}e^{-rt}\nu(\mathrm{d}t).$$

2. 2.

   Non-isotropic fields with isotropic increments. The structure function $D_{N}$ can be written as

   $\displaystyle D_{N}(r)=\int_{(0,\infty)}\left(1-\Lambda_{N}(\sqrt{rt})\right)\nu_{N}(\mathrm{d}t)+A_{N}r,$

   (1.3)

   where $A_{N}\in\mathbb{R}_{+}$ is a constant and $\nu_{N}$ is a $\sigma$-finite measure with

   $$\int_{(0,\infty)}\frac{t}{1+t}\nu_{N}(\mathrm{d}t)<\infty.$$

   For $N=\infty$, the structure function $D(r):=D_{\infty}(r)$ has the following form

   $\displaystyle D(r)=\int_{(0,\infty)}\left(1-e^{-rt}\right)\nu(\mathrm{d}t)+Ar,$

   (1.4)

   which is, in the language of Theorem 3.2 below, a Bernstein function with $D(0)=0$.

These representations provide us some information on the sign of derivatives provided they are finite. For instance, \reftagform@1.4 implies that $D^{\prime}(r)\geq 0$ and $D^{\prime\prime}(r)\leq 0$ for $r\geq 0$. However, due to the oscillatory nature of Bessel functions, we only have $D_{N}^{\prime}(0)\geq 0$ and $D_{N}^{\prime\prime}(0)\leq 0$ for $N\in\mathbb{N}$.

Let $E\subset\mathbb{R}$ and $T_{N}\subset\mathbb{R}^{N}$ be Borel subsets. If the random field is twice differentiable, we define

where $i(\nabla^{2}H_{N}(x))$ is the index (or the number of negative eigenvalues) of the Hessian $\nabla^{2}H_{N}(x)$. Under some suitable smoothness conditions in [AT07], Cheng and Schwartzman gave a representation of $\mathbb{E}[\mathrm{Crt}_{N,k}([u,\infty),T)]$ for the isotropic fields using the Kac–Rice formula in [CS18, Theorem 3.5], where $u\in[-\infty,\infty)$ and $T$ is the unit-volume ball in $\mathbb{R}^{N}$. A key ingredient in the representation is what the authors call the Gaussian Orthogonally Invariant (GOI) matrix, which was first studied by Mallows [Ma61]. Following their terminology, an $N\times N$ real symmetric random matrix $M$ is said to be Gaussian Orthogonally Invariant (GOI) with covariance parameter $c$, denoted by $\operatorname{GOI}(c)$, if its entries $M_{ij},1\leq i,j\leq N$ are centered Gaussian random variables such that

Note that the distribution of such matrices is invariant under the congruence transformation by any orthogonal matrix. Therefore, they share some properties of the GOE matrices. In particular, for $c=0$, GOI$(0)$ matrices are exactly GOE matrices. Recall that a Gaussian vector is nondegenerate if and only if the covariance matrix of the Gaussian vector is positive definite. Cheng and Schwartzman [CS18] showed that a GOI$(c)$ matrix of size $N\times N$ is nondegenerate if and only if $c>-\frac{1}{N}$. Moreover, for this nondegenerate GOI$(c)$ matrix, they derived that the density of the ordered eigenvalues is given by

where $K_{N}=2^{N/2}\prod_{i=1}^{N}\Gamma\left(\frac{i}{2}\right)$ is the normalization constant. For $c\neq 0$, one may write a GOI$(c)$ matrix as a GOE matrix plus a random scalar matrix. If $c>0$ the scalar matrix is independent of the GOE matrix, while if $c<0$ the scalar matrix is no longer independent. In the limit $N\to\infty$ or equivalently for structure functions $D\in\mathcal{D}_{\infty}$, the covariance parameter $c$ is positive and thus for these structure functions and all dimensions $N\in\mathbb{N}$, the Kac–Rice representation of $\mathbb{E}[\mathrm{Crt}_{N,k}(E,T_{N})]$ can always be given by GOE matrices. This was the setting in the pioneering works of Fyodorov and Nadal [Fy04, FN12] where the GOE matrices are crucial for asymptotic analysis. However, if one insists on considering structure functions from $\mathcal{D}_{N}$ for fixed $N$, then additional restriction is needed in order to get a GOE matrix (plus an independent scalar matrix) in the Kac–Rice representation.

Before illustrating similar problems for the non-isotropic Gaussian fields, let us first put a remark on the relationship between the two cases. In general, the two types of Gaussian random fields are quite different, like the Ornstein–Uhlenbeck process and Brownian motion in dimension $1$. But if $\nu_{N}$ is a finite measure in the second case, the non-isotropic field is essentially a shifted isotropic field. Indeed, let $H_{N}(x)$ be an isotropic field that satisfies

Then we can verify that $\widehat{H}_{N}(x):=H_{N}(x)-H_{N}(0)$ satisfies

and $\widehat{H}_{N}(0)=0$, which means that $\widehat{H}_{N}(x)$ is a non-isotropic Gaussian random field with isotropic increments. Conversely, let $\widehat{H}_{N}(x)$ be a non-isotropic field satisfying \reftagform@1.6 with some finite measure $\nu_{N}$. Define $H_{N}(x)=\widehat{H}_{N}(x)+H_{N}(0)$, where $H_{N}(0)$ is a centered Gaussian random variable with variance $\mathbb{E}[H_{N}(0)^{2}]=\frac{1}{2}|\nu_{N}|$ and

One can check that $H_{N}(x)$ is an isotropic Gaussian random field. Due to the relation $H_{N}(x)-H_{N}(0)=\widehat{H}_{N}(x)$, the non-isotropic field $\widehat{H}_{N}(x)$ has the same critical points and landscape as those of the isotropic field $H_{N}$. Furthermore, let $\widetilde{H}_{N}(x)$ be a non-isotropic Gaussian random field satisfying

with some finite measure $\nu_{N}$ and constant $A_{N}>0$. In this case, we can split $\widetilde{H}_{N}(x)$ into two independent non-isotropic parts $\widetilde{H}_{N}(x)=\widehat{H}_{N}(x)+\overline{H}_{N}(x)$, where $\widehat{H}_{N}(x)$ satisfies \reftagform@1.6 and $\overline{H}_{N}(x)$ satisfies

Note that we may write $\overline{H}_{N}(x)=\langle x,\xi\rangle$ for a centered Gaussian random vector $\xi=(\xi_{1},\xi_{2},\dots,\xi_{N})$ with covariance matrix $A_{N}\mathbf{I}_{N}$, where $\langle\cdot,\cdot\rangle$ is the Euclidean inner product and $\mathbf{I}_{N}$ denotes the $N\times N$ identity matrix hereafter. In this case, the field $\overline{H}_{N}$ is almost trivial since its gradient is a fixed random vector and we have $\nabla\widetilde{H}_{N}(x)=0$ if and only if $\nabla\widehat{H}_{N}(x)=-\xi$. Therefore, when we talk about the non-isotropic fields, we may assume that $\lim_{r\to\infty}D(r)=\infty$ and the measure $\nu_{N}$ is $\sigma$-finite but not finite.

For the non-isotropic fields with isotropic increments, it is expected that the above representation problem is more challenging since the distribution of the field $H_{N}(x)$ depends on the location variable $x\in\mathbb{R}^{N}$. Recently, Auffinger and Zeng considered the Kac–Rice representation of $\mathbb{E}[\operatorname{Crt}_{N,k}\left(E,T_{N}\right)]$ for the non-isotropic Gaussian fields with isotropic increments in the limit $N\to\infty$ during their study of the landscape complexity in [AZ20, AZ22], where GOE matrices are indispensable in the analysis as the isotropic case. This left the representation problem open for the fixed finite dimensions. Moreover, in order to utilize GOE matrices, they assumed a technical condition (see Assumption III below) and verified it for some special cases or subclass of $\mathcal{D}_{\infty}$. They conjectured that this condition always holds for all structure functions belonging to $\mathcal{D}_{\infty}$.

This paper aims to resolve these issues for the non-isotropic Gaussian fields with isotropic increments. In Section 2, we prove the main results of this paper, which display various representations of $\mathbb{E}[\operatorname{Crt}_{N,k}\left(E,T_{N}\right)]$. The basic tool is the Kac–Rice formula as usual. In general, the representation can be obtained by using GOI$(c)$ matrices (see Theorem 2.6). Furthermore, we may reduce the GOI$(c)$ matrices to GOE matrices in the representation, in the special case of $E=\mathbb{R}$ (see Theorem 2.7) or with Assumption III (see Theorem 2.10). These results can be regarded as the non-isotropic analog of those for the isotropic Gaussian fields in [CS18]. In Section 3, we show that for the structure functions in $\mathcal{D}_{\infty}$, $\mathbb{E}[\operatorname{Crt}_{N,k}(E,T_{N})]$ can always be represented by using GOE matrices when $T_{N}$ is a shell domain. In other words, the aforementioned technical condition is redundant as conjectured in [AZ20].