No repulsion between critical points for planar Gaussian random fields

We study the behaviour the point process of critical points of isotropic stationary Gaussian fields. We compute the main term in the asymptotic expansion of the two-point correlation function near the diagonal. Our main result could be interpreted as a statement that for a 'generic' field the critical points neither repel no attract each other. Our analysis also allows to study how the short-range behaviour of critical points depends on their index.


Introduction
1.1. Two-point correlation function for critical points of planar random fields. The number of critical points of a function and their positions are its important qualitative descriptor, and their study is an actively pursued field of research within a wide range of disciplines, such as classical analysis (see e.g. [8]), probability (e.g. [9,10]), mathematical and theoretical physics ( [7]), spectral geometry (e.g. [13,11]), and cosmology and the study of Cosmic Microwave Background (CMB) radiation (e.g [6]). In case F : R 2 → R (or, more generally, F : R d → R, d ≥ 2) is a smooth Gaussian random field, then its set of critical points C F is a point process on R 2 (resp. R d ). If we assume in addition that F is stationary, then it is possible to employ the Kac-Rice method in order to obtain that, under some mild non-degeneracy assumptions on F and its mixed derivatives up to 2nd order, the expected number of critical points lying in a ball B(R) ⊆ R 2 is given precisely by where c F > 0 is a constant that could be expressed in terms of some derivatives of the covariance function of F evaluated at the diagonal.
It is then compelling to study the law of C F in more depth, e.g., the variance of For C F (and other point processes that are zeros of random Gaussian fields, with C F being the zero set of ∇F ), one can usually derive the 2-point correlation function via the Kac-Rice formula K F 2 (x, y) = φ (∇F (x),∇F (y)) (0, 0) · E |det H F (x) · det H F (y)| ∇F (x) = ∇F (y) = 0 , (1.2) where φ (∇F (x),∇F (y)) (·) is the Gaussian density of the vector (∇F (x), ∇F (y)) ∈ R 2 × R 2 , and H F (·) is the Hessian of F . The function (1.2) is, in turn, a semi-explicit function of the covariance function of F and its couple of mixed derivatives.
If, in addition, F is assumed to be isotropic, then K F 2 (x, y) is a function of the Euclidean distance r = x−y . In many cases, when the covariance function of F is decaying sufficiently rapidly, the long range asymptotics of K F 2 (r), r → ∞ yields the asymptotic variance of the number of critical points in large balls B(R), R → ∞, see e.g. [9,10], and other quantities, such as the nodal length of F [4,14]. On the contrary, the short range asymptotics of K F 2 (r), r → 0 yields the asymptotic law of the second factorial moment of the number of critical points of F belonging to small balls B(r), r → 0, again via (1.1). Informally, the probability that there is one critical point in a ball B(r) of small radius r > 0 is approximately cr 2 , whereas the probability that there are two critical points in B(r) is approximately B(r)×B(r) K 2 (x, y)dxdy. If K 2 (r) → ∞ as r → 0, then the probability to have two critical points in B(r) is much higher than the square of the probability to have one critical point in B(r). In this case we say that the critical points attract each other. Otherwise, if K 2 (r) → 0 as r → 0, then the probability to have two critical points in B(r) is much lower than the square of the probability to have one such point. In this case we say that the critical points repel each other.
The first relevant result [3] was obtained in 2017 when we analysed the asymptotic behaviour of K 2 for a particular Gaussian field: the random monochromatic isotropic plane waves, also referred to as "Berry's Random Wave Model" (RWM). This field is of a particular interest since it is believed to represent the (deterministic) Laplace eigenfunctions on "generic" chaotic surfaces, in the high energy limit [5]. The RWM is the stationary isotropic random field F : R 2 → R, uniquely defined by the covariance function The work [3] was motivated by Figure 1 (left) which apparently shows that the critical point repel each other. It was found [3, p. 10] that for RWM the two-point correlation function is asymptotic around the diagonal to so that, in particular, the critical points of F exhibit no repulsion nor attraction. It was then inferred that the seemingly visible repulsion on some numerically generated pictures could be attributed to rigidity of critical points, a notion cardinally different from repulsion. The work [3] also allowed for the separation of the critical points into maxima, minima and saddles, and studied the effect of such a separation on the expansion of the corresponding 2-point correlation function, resulting in some cases in qualitatively di↵erent behaviour to (1.3). It is then natural to inquire about the analogous question for other Gaussian random fields, i.e. for the asymptotic law of the 2-point-correlation function around the diagonal for other Gaussian random fields. In particular, whether it is true, that for a generic stationary field, the critical points nether attract nor repel each other. That critical points do not attract was resolved by [12], who, among other things, proved that for 'generic' stationary planar Gaussian random fields, without assuming that the underlying random field is isotropic.
Our first principal Theorem 1.2 below yields that for a Gaussian isotropic random field F satisfying some generic assumptions, one has with a F > 0 explicitly evaluated in terms of some derivatives of the covariance function of F . In particular, it implies that the critical points of an isotropic stationary Gaussian field do not repel each other (see Corollary 1.4 below). The value of a F in (1.4) is of no particular significance other than its mere positivity. What might be of some interest is the relation between the density of the critical points c F and a F . It can show, by comparison with a Poisson process of the same intensity c F , that critical pints are more clustered or dispersed on a small scale compared to the corresponding Poisson process. Finally, we do believe that the isotropic assumption is not essential for the validity of (1.4).

Statement of the main results.
Our main result concerns the short range asymptotics for the 2-point correlation function corresponding to smooth stationary isotropic Gaussian fields F : be the covariance function of F . Assuming that F is su ciently smooth and unit variance, and taking into account that C F is even and for every k 0, , we may Taylor expand C F (r) around the origin as C F (r) = 1 g 2 r 2 + g 4 r 4 g 6 r 6 + O(r 8 ), (1.5) NO REPULSION BETWEEN CRITICAL POINTS 3 Figure 1. Critical points for the random plane wave (left), Bargmann-Fock field (center) and an anisotropic field. Minima and maxima are red and blue plusses and crosses, critical points are black dots.
2-point correlation function, resulting in some cases in qualitatively di↵erent behaviour to (1.3). It is then natural to inquire about the analogous question for other Gaussian random fields, i.e. for the asymptotic law of the 2-point-correlation function around the diagonal for other Gaussian random fields. In particular, whether it is true, that for a generic stationary field, the critical points nether attract nor repel each other. That critical points do not attract was resolved by [12], who, among other things, proved that for 'generic' stationary planar Gaussian random fields, without assuming that the underlying random field is isotropic.
Our first principal Theorem 1.2 below yields that for a Gaussian isotropic random field F satisfying some generic assumptions, one has with a F > 0 explicitly evaluated in terms of some derivatives of the covariance function of F . In particular, it implies that the critical points of an isotropic stationary Gaussian field do not repel each other (see Corollary 1.4 below). The value of a F in (1.4) is of no particular significance other than its mere positivity. What might be of some interest is the relation between the density of the critical points c F and a F . It can show, by comparison with a Poisson process of the same intensity c F , that critical pints are more clustered or dispersed on a small scale compared to the corresponding Poisson process. Finally, we do believe that the isotropic assumption is not essential for the validity of (1.4).

Statement of the main results.
Our main result concerns the short range asymptotics for the 2-point correlation function corresponding to smooth stationary isotropic Gaussian fields F : be the covariance function of F . Assuming that F is su ciently smooth and unit variance, and taking into account that C F is even and for every k 0, , we may Taylor expand C F (r) around the origin as C F (r) = 1 g 2 r 2 + g 4 r 4 g 6 r 6 + O(r 8 ), (1.5) NO REPULSION BETWEEN CRITICAL POINTS 3 Figure 1. Critical points for the random plane wave (left), Bargmann-Fock field (center) and an anisotropic field. Minima and maxima are red and blue plusses and crosses, critical points are black dots.

2-point correlation function, resulting in some cases in qualitatively di↵erent behaviour to (1.3).
It is then natural to inquire about the analogous question for other Gaussian random fields, i.e. for the asymptotic law of the 2-point-correlation function around the diagonal for other Gaussian random fields. In particular, whether it is true, that for a generic stationary field, the critical points nether attract nor repel each other. That critical points do not attract was resolved by [12], who, among other things, proved that for 'generic' stationary planar Gaussian random fields, without assuming that the underlying random field is isotropic.
Our first principal Theorem 1.2 below yields that for a Gaussian isotropic random field F satisfying some generic assumptions, one has with a F > 0 explicitly evaluated in terms of some derivatives of the covariance function of F . In particular, it implies that the critical points of an isotropic stationary Gaussian field do not repel each other (see Corollary 1.4 below). The value of a F in (1.4) is of no particular significance other than its mere positivity. What might be of some interest is the relation between the density of the critical points c F and a F . It can show, by comparison with a Poisson process of the same intensity c F , that critical pints are more clustered or dispersed on a small scale compared to the corresponding Poisson process. Finally, we do believe that the isotropic assumption is not essential for the validity of (1.4).

Statement of the main results.
Our main result concerns the short range asymptotics for the 2-point correlation function corresponding to smooth stationary isotropic Gaussian fields F : be the covariance function of F . Assuming that F is su ciently smooth and unit variance, and taking into account that C F is even and for every k 0, , we may Taylor expand C F (r) around the origin as C F (r) = 1 g 2 r 2 + g 4 r 4 g 6 r 6 + O(r 8 ), (1.5) It is then natural to inquire about the analogous question for other Gaussian random fields, i.e. for the asymptotic law of the 2-point-correlation function around the diagonal for other Gaussian random fields. In particular, whether it is true, that for a generic stationary field, the critical points nether attract nor repel each other. That critical points do not attract was resolved by [12], who, among other things, proved that for 'generic' stationary planar Gaussian random fields, without assuming that the underlying random field is isotropic.
Our first principal Theorem 1.2 below yields that for a Gaussian isotropic random field F satisfying some generic assumptions, one has with a F > 0 explicitly evaluated in terms of some derivatives of the covariance function of F . In particular, it implies that the critical points of an isotropic stationary Gaussian field do not repel each other (see Corollary 1.4 below). The value of a F in (1.4) is of no particular significance other than its mere positivity. What might be of some interest is the relation between the density of the critical points c F and a F . It can show, by comparison with a Poisson process of the same intensity c F , that critical pints are more clustered or dispersed on a small scale compared to the corresponding Poisson process. Finally, we do believe that the isotropic assumption is not essential for the validity of (1.4).

Statement of the main results.
Our main result concerns the short range asymptotics for the 2-point correlation function corresponding to smooth stationary isotropic Gaussian fields F : be the covariance function of F . Assuming that F is sufficiently smooth and unit variance, and taking into account that C F is even and for every k ≥ 0, , we may Taylor expand C F (r) around the origin as where for all k ≥ 1, we have By rescaling F if necessary, we may further assume w.l.o.g. that g 2 = 1 (if g 2 vanishes it would force F to be a.s. linear, which would contradict F being isotropic, unless F is constant), so that (1.5) reads (1.7) To simplify the formulas in the main theorem we introduce the following notation: (1.9) Theorem 1.2. Let F : R 2 → R be a nonconstant stationary isotropic Gaussian random field, and assume that F is a.s. C 3+ (R 2 ). Then the 2-point correlation function corresponding to the critical points admits the following expansion around the origin: where g 2k are given by (1.6), and φ(·, ·), A(·, ·) and B(·, ·) are given by ( We would like to analyse the leading term in (1.10), in particular, whether it may vanish for some values of g 4 , g 6 that do correspond to some random field, equivalently, whether A(g 4 , g 6 )+B(g 4 , g 6 ) might vanish. We will show below (see (3.11)) that A(g 4 , g 6 )+B(g 4 , g 6 ) = 0 for some real strictly positive g 4 , g 6 , if and only if To analyse this equation we write the derivatives in terms of the spectral measure ρ of F (that is, ρ is the Fourier transform of the covariance kernel C(·) on R 2 ): By the Cauchy-Schwarz inequality and formulas (1.6) and (1.11) we obtain the following inequality between g 4 and g 6 : The equality holds if and only if ρ is the δ-measure at the origin, equivalently, F is a (random) constant. This means that the leading term is non-zero for non-degenerate F . Our next result is analogous to Theorem 1.2, while separating the critical points into different types: minima, maxima and saddle points.

Expected Number of Critical Points
Counting the critical points in a ball B(R) ⊆ R 2 is equivalent to counting the zeros of the map x → ∇F (x). By the Kac-Rice formula the density of critical points is where φ ∇F (x) is the Gaussian probability density of two-dimensional vector ∇F (x) ∈ R 2 evaluated at 0. By the Kac-Rice formula, if ∇F (x) is nonsingular for all x ∈ B(R), then where in the last step we use the fact that F is assumed isotropic. To write an analytic expressions for K 1 we evaluate the covariance matrix Σ (see Appendix A) of the 5-dimensional centred jointly Gaussian vector (∇F (x), ∇ 2 F (x)) where ∇ 2 F (x) is the vectorized Hessian evaluated at x: Now using the value of the matrix A, and thanks to the statistical independence of the first and the second order mixed derivatives of F at every fixed point x ∈ R 2 , we have Using the value of the covariance matrix C of ∇ 2 F (x), and following the argument in the proof of [10, Proposition 1.1], we note that

is a centred jointly Gaussian random vector with covariance matrix
The statement follows combining (2.1), (2.2) and (2.3): 3. Second Factorial Moment 3.1. On the Kac-Rice formula for computing the second factorial moment of the number of critical points.
As explained in the introduction, i.e., a Gaussian integral involving the covariance function C F and its derivatives. This naturally reduces to studying the distribution of the centred Gaussian vector with covariance matrix Σ(x, y), x, y ∈ B(r). It is known [2, Theorem 6.9] that, if for all x = y the Gaussian distribution of (∇F (x), ∇F (y)) is non-degenerate, the second factorial moment of the number of critical points in B(r) can be expressed as We note that K 2 is everywhere nonnegative.

Proof of Theorem 1.2.
Proof. In order to study the asymptotic behaviour of the second factorial moment of the number of critical points in B(r), as the radius r of the disk goes to zero, we need to study the centred Gaussian random vector (3.1). Its covariance matrix Σ = Σ(x, y) is of the form where A = A(x, y) is the covariance matrix of the gradients (∇F (x), ∇F (y)), C = C(x, y) is the covariance matrix of the second order derivatives (∇ 2 F (x), ∇ 2 F (y)) and B = B(x, y) is the covariance matrix of the first and second order derivatives. The function F is isotropic, hence, the law of the critical point process is also invariant w.r.t. translations and rotations. This means that its 2-point function K 2 (x, y) depends on ||x − y|| only (but not the covariance matrix Σ); by the standard abuse of notation we write We will asymptotically evaluate K 2 (x, y) for x = (0, 0) and y = (0, r) in the relevant regime, which, thanks to the by-product (3.3) of the isotropic property of F , will also yield the same for K 2 (r).
In Appendix B we evaluate the entries of Σ(x, y) for the said x and y, and in Appendix C we evaluate the covariance matrix ∆ = ∆(x, y) of (∇ 2 F (x), ∇ 2 F (y)) conditioned on ∇F (x) = ∇F (y) = 0, i.e., From now on we will only work with Σ(r) and ∆(r) are defined (not canonically) as Σ(x, y) and ∆(x, y) with x = (0, 0) and y = (0, r).
Our aim is to study the asymptotic behaviour of the 2-point correlation function K 2 in the vicinity of r = 0. When facing an integral of this type, it is useful to transform the coordinates so that rewrite the integrand in terms of the standard Gaussian vector, as below.
The equality (3.5) implies that we can write This suggests to introduce a new variable ξ = Λ −1/2 (r)P(r)ζ. Clearly, we can express ζ in terms of ξ as ζ = P −1 (r)Λ 1/2 (r)ξ = P t (r)Λ 1/2 (r)ξ (3.7) With this transformation of variables 1 and upon using (3.7), we can express the components ζ i as where the q ij (r) are the elements of Q(r) = P −1 (r) = P t (r). The columns of Q form an orthonormal basis of the eigenvectors of ∆(r). With this transformation of variables we can rewrite the two quadratic forms ζ 1 ζ 3 − ζ 2 2 and ζ 4 ζ 6 − ζ 2 5 in (3.4) as (3.8) Substiting (3.6) into (3.4) we get where ζ 1 ζ 3 − ζ 2 2 and ζ 4 ζ 6 − ζ 2 5 are functions of ξ i as described by (3.8). To obtain the asymptotic behaviour around r = 0 of the integral in (3.9), we Taylor expand around the origin all terms in (3.8). For this we need the first few terms of the Taylor expansion of q ij and λ j . In Lemmas C.1 and C.2 we express them as functions of the entries of the matrix ∆. The entries of ∆ are given in terms of entries of Σ which are given in terms of the derivatives of the covariance kernel C (see Appendices A and B). Expanding C as in (1.7) we obtain (with the aid of symbolic computations in Mathematica) first few terms of the Taylor expansion of the integrand from (3.9) and the determinant of A. The result are given below. The matrix A has a simple block structure and it is easy to compute its determinant. An explicit computation in Appendix B gives det(A(r)) = 16 √ 3g 4 r 2 − 32 √ 3(g 2 4 + g 6 )r 4 + O(r 6 ). For the product of expressions from (3.8) we have where φ(g 4 , g 6 ), ϕ(g 4 , g 6 ), A(g 4 , g 6 ), and B(g 4 , g 6 ) are defined in (1.8) and (1.9). Combining these expansions (3.10) The multiple integral in (3.10) can be written as a product of one-dimensional integral. Using a standard fact that we can rewrite (3.10) as We finally obtain that, as r → 0, and, in view of (3.2), as r → 0, We observe that, under the condition g 4 , g 6 > 0, we have that φ(g 4 , g 6 ) > 0. We also note that A(g 4 , g 6 ) + B(g 4 , g 6 ) = −(4 g 2 4 − 10 g 6 ) φ(g 4 , g 6 ), (3.11) so A(g 4 , g 6 ) + B(g 4 , g 6 ) = 0, if and only if

Proof of Theorem 1.5: minima, maxima and saddles
To prove Theorem 1.5 we need to evaluate the two-point correlation function K 2 modified for the respective types of critical points. The modified function K 2 has the same expression (3.4) with the integration over a proper subset of R 6 , that is the ζ are restricted to a domain corresponding to the prescribed type of critical points.
Let us introduce two Hessians at points x and y (already conditioned to be critical points). In terms of ζ i these Hessians are given by The particular type of a critical point depends on the eigenvalues of its Hessian, we may reformulate this dependency in terms of the following quantities: a critical point with Hessian H i is a minimum if c i > 0 and b i < 0, a maximum if c i > 0 and b i > 0, and a saddle if c i < 0 (we ignore the probability 0 event when one of the eigenvalues vanishes). As before, we rewrite ζ i in terms of ξ i . Expanding in powers of r we get We observe that all the coefficients b i,j are linear functions of the coordinates of ξ, and all the coefficients c i,j are quadratic forms of the entries of ξ, and also notice that ξ 6 ξ 4 B(g 4 , g 6 ) + ξ 3 A(g 4 , g 6 ) Note that φ(g 4 , g 6 ) > 0 (unless g 4 = g 6 = 0) and that B(g 4 , g 6 ) ≥ 0 (given g 2 4 < 5 2 g 6 ) and equal to zero if and only if g 4 = g 6 = 0. We introduce s i = ξ i /|ξ|, s ∈ S 5 , |ξ| ∈ (0, ∞), abusing notation we denote with the same letters the rescaled coefficients that are now function of s i instead of ξ i . With this notation, (3.9) reads K 2 (r) = 12 where c j are as in (4.1), and ds is the spherical volume element on the unit sphere S 5 .

4.1.
Minimum-minimum. The two-point correlation function K min,min 2 (r) corresponding to the local minima is given by (4.3) with integration domain Since c 1,1 = −c 2,1 , for some constant C sufficiently big we have In the case b 1,1 = 0 (which is the case for RWM), the condition that b 1 and b 2 are of different sign implies that |s 6 | < Cr 2 for some big constant C. Under the assumption |s 6 | < Cr 2 both b i are of the form and again, b i of different signs, forces the term corresponding to O(r 3 ) to dominate, that is Combining all of this we obtain the estimate Appendix A. Covariance matrix of (∇F (x), ∇ 2 F (x)) By translation invariance, the covariance matrix Σ of the 5-dimensional centred Gaussian vector which combines the gradient and the vectorized Hessian evaluated at x, does not depend on the point x ∈ R 2 . It is convenient to write Σ as a block matrix: The computations of A, B and C do not require sophisticated arguments other than iterative differentiation of the covariance function C F (||x−y||), with covariances of the derivative given by derivatives of the covariance kernel. As a concrete example of such computation here are the details of the computation for (A) 1,1 . Taking into account the Taylor expansion of C F (r) around the origin: C F (||x − y||) = 1 − x − y 2 + g 4 ||x − y|| 4 − g 6 ||x − y|| 6 + g 8 ||x − y|| 8 + O(||x − y|| 10 ), we have With analogous calculations, but using higher order derivatives of C F (||x − y||) we compute the entries of C. Finally, since the first and second order derivatives of any stationary field are independent at every fixed point x ∈ R 2 , we immediately have B = 0.
Appendix B. Covariance matrix of (∇F (x), ∇F (x), ∇ 2 F (x), ∇ 2 F (y)) In this section we compute the covariance matrix Σ(x, y) for the 10-dimensional Gaussian random vector which combines the gradient and the vectorized Hessian evaluated at x = (0, 0) and y = (0, r); thanks to the isotropic property of the field F , this is sufficient in order to evaluate K 2 (r) for all relevant r. It is convenient to write the matrix Σ(z, w) in block form, that is Σ(x, y) = A(x, y) B(x, y) B t (x, y) C(x, y) , Σ(x, y)| x=(0,0),y=(0,r) = Σ(r).
Analogously to the above, we derive the entries of the matrices B(r) and C(r):