Critical point asymptotics for Gaussian random waves with densities of any Sobolev regularity

TL;DR

This paper investigates how the regularity of Gaussian random waves affects the asymptotic growth of their critical points, revealing a transition from area to diameter growth depending on the Sobolev regularity parameter.

Contribution

It establishes a detailed connection between Fourier transform regularity and critical point asymptotics, including phase transitions and precise asymptotic expansions for Bessel series.

Findings

Critical points grow like area for low regularity ($s<1.5$).

Critical points grow like diameter for high regularity ($s>2.5$).

At intermediate regularity, growth transitions linearly with radius.

Abstract

We consider Gaussian random monochromatic waves $u$ on the plane depending on a real parameter $s$ that is directly related to the regularity of its Fourier transform. Specifically, the Fourier transform of $u$ is $f\,d\sigma$, where $d\sigma$ is the Hausdorff measure on the unit circle and the density $f$ is a function on the circle that, roughly speaking, has exactly $s-\frac12$ derivatives in $L^2$ almost surely. When $s=0$, one recovers the classical setting for random waves with a translation-invariant covariance-kernel. The main thrust of this paper is to explore the connection between the regularity parameter $s$ and the asymptotic behavior of the number $N(\nabla u,R)$ of critical points that are contained in the disk of radius $R\gg1$. More precisely, we show that the expectation $\mathbb{E}N(\nabla u,R)$ grows like the area of the disk when the regularity is low enough ($s<\frac32$) and like the diameter when the regularity is high enough ($s>\frac52$), and that the corresponding exponent changes according to a linear interpolation law in the intermediate regime. The transitions occurring at the endpoint cases involve the square root of the logarithm of the radius. Interestingly, the highest asymptotic growth rate occurs only in the classical translation-invariant setting, $s=0$. A key step of the proof of this result is the obtention of precise asymptotic expansions for certain Neumann series of Bessel functions. When the regularity parameter is $s>5$, we show that in fact $N(\nabla u,R)$ grows like the diameter with probability 1, albeit the ratio is not a universal constant but a random variable.