Two Point Function for Critical Points of a Random Plane Wave

Abstract

Random plane wave is conjectured to be a universal model for high-energy eigenfunctions of the Laplace operator on generic compact Riemanian manifolds. This is known to be true on average. In the present paper we discuss one of important geometric observable: critical points. We first compute one-point function for the critical point process, in particular we compute the expected number of critical points inside any open set. After that we compute the short-range asymptotic behaviour of the two-point function. This gives an unexpected result that the second factorial moment of the number of critical points in a small disc scales as the fourth power of the radius.∞ n=−∞ a n J |n| (k r)e inθ in polar coordinates, where J n are Bessel functions and a n are independent complex Gaussian random variables with variance 2. Since the Bessel functions decay exponentially fast as functions

A second moment bound for critical points of planar Gaussian fields in shrinking height windows

“…Since by Gaussian regression these diagonal elements are E[(∇ 2 f (x)) 2 i,j | (∇f (x) = ∇f (y) = 0)] for (i, j) = (1, 1), (1,2), (2, 2) respectively, we deduce that N (r) = O(r 2 ) as claimed. Finally, by Gaussian regression, σ 2 1 (x, y) = det(Σ 4 (x, y))/det(Σ 2 (x, y)), and since, by Lemma 3.3,…”

“…Since ∂ (4,0) K x = E[(∂ (2,0) f (x)) 2 ] and ∂ (2,2) K x = E[(∂ (1,1) f (x)) 2 ], the claimed asymptotics for det(Σ 4 (x, y)) follow from (2.5). Again applying Lemma 3.3, the diagonal elements of Σ 5 (x, y) T Σ 2 (x, y) −1 Σ 5 (x, y) are equal, respectively, to…”

“…Conditions 1.1 and 1.2 are extremely mild, and will be satisfied in most applications of interest. Notably, while these conditions imply that f (0) and ∇ 2 f (0) are non-degenerate (ignoring the trivial degeneracy in the mixed partial derivatives), we do not insist that (f (0), ∇ 2 f (0)) be jointly non-degenerate (as in [10] for instance), and so they hold in particular for the 'random plane wave' (the case κ(x) = J 0 (|x|), where J 0 is the zeroth Bessel function; see, e.g., [2,5,15]).…”

Anisotropic Gaussian wave models

“…Another mathematical pioneer study was raised by Berry in several papers, Berry (1977) or Berry and Dennis (2000) for instance. These seminal works opened a wide area of research in the last decades, either for statistical purposes (Azaïs et al, 2005, Lindgren, 2006, Aberg et al, 2008, Azaïs et al, 2011, Beliaev et al, 2019, Nourdin et al, 2019, or more recently for topological purposes in link with number theory (Rudnick and Wigman, 2008, Krishnapur et al, 2013, Marinucci et al, 2016. Ten years ago, the interest for nodal sets or level sets also met the theory of crossings developed by Rice for one-dimensional stochastic processes fifty years before, yielding two inspiring books by Adler and Taylor (2007) and by Azaïs and Wschebor (2009).…”

“…In order to illustrate our results, some models are specified. A famous case is Berry's monochromatic random waves model (see Berry, 1977 as a precursor or Berry, 2002;Beliaev et al, 2019;Nourdin et al, 2019 for recent studies in a twodimensional context), which is the isotropic solution of Helmholtz equation and is prescribed by a random wavevector uniformly distributed on a sphere. The planar version of Berry's monochromatic random waves has been extensively studied in the last decade for the outstanding behaviour of its nodal length variance in the highenergy limit.…”