2021/07/07 by Alberto Enciso, Daniel Peralta‐Salas, Enciso, Alberto +3
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometry and complex manifolds #Probability (math.PR) #Spectral Theory (math.SP)
paper · pdf · doi:10.48550/arxiv.2107.03363
openalex publication_date 2021/07/07 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
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 dσ, where dσ 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 L2 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(∇ u,R) of critical points that are contained in the disk of radius R≫1. More precisely, we show that the expectation 𝔼N(∇ u,R) grows like the area of the disk when the regularity is low 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(∇ u,R) grows like the diameter with probability 1, albeit the ratio is not a universal constant but a random variable.