Hitting probabilities of Gaussian random fields and collision of eigenvalues of random matrices

TL;DR

This paper establishes a new sufficient condition for the polar nature of sets for Gaussian random fields, extending previous results, and applies it to prove the existence of eigenvalue collisions in Gaussian matrix ensembles.

Contribution

It provides a significantly improved criterion for the polar sets of Gaussian fields and solves an open problem on eigenvalue collisions in Gaussian random matrices.

Findings

New sufficient condition for sets to be polar for Gaussian fields

Extension of previous results from singleton to more general sets

Proof of eigenvalue collision existence in Gaussian random matrices

Abstract

Let $X= \{X(t), t \in \mathbb R^N\}$ be a centered Gaussian random field with values in $\mathbb R^d$ satisfying certain conditions and let $F \subset \mathbb R^d$ be a Borel set. In our main theorem, we provide a sufficient condition for $F$ to be polar for $X$, i.e. $\mathbb P \big( X(t) \in F \hbox{ for some } t \in \mathbb R^N \big) = 0$, which improves significantly the main result in Dalang et al [7], where the case of $F$ being a singleton was considered. We provide a variety of examples of Gaussian random field for which our result is applicable. Moreover, by using our main theorem, we solve a problem on the existence of collisions of the eigenvalues of random matrices with Gaussian random field entries that was left open in Jaramillo and Nualart [14] and Song et al [21].