Asymptotic topology of excursion and nodal sets of Gaussian random fields

TL;DR

This paper investigates the asymptotic topological structure of high-level excursion and nodal sets of Gaussian random fields on manifolds, revealing they are predominantly composed of simple geometric shapes like balls and spheres.

Contribution

It introduces a Morse theory-based method to analyze the expected topology of excursion and nodal sets of Gaussian fields, providing new asymptotic formulas and extending to spherical spin glasses.

Findings

Expected number of topological n-balls in excursion sets asymptotically computed

High nodal sets are mostly spheres with similar asymptotics as excursion sets

Asymptotics of Nazarov-Sodin constant derived for large thresholds

Abstract

Let M be a compact smooth manifold of dimension n with or without boundary, and f : M $\rightarrow$ R be a smooth Gaussian random field. It is very natural to suppose that for a large positive real u, the random excursion set {f $\ge$ u} is mostly composed of a union of disjoint topological n-balls. Using the constructive part of (stratified) Morse theory we prove that in average, this intuition is true, and provide for large u the asymptotic of the expected number of such balls, and so of connected components of {f $\ge$ u}, see Theorem 1.2. We similarly show that in average, the high nodal sets {f = u} are mostly composed of spheres, with the same asymptotic than the one for excursion set. A refinement of these results using the average of the Euler characteristic given by [2] provides a striking asymptotic of the constant defined by F. Nazarov and M. Sodin, again for large u, see Theorem 1.11. This new Morse theoretical approach of random topology also applies to spherical spin glasses with large dimension, see Theorem 1.14.