Hitting probabilities of Gaussian random fields and collision of eigenvalues of 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. $\mathbb P\big ( X(t) \in F \text { for some } t \in \mathbb {R}^N\big ) = 0$, which improves significantly the main result in Dalang et al. [Ann. Probab. 45 (2017), pp. 4700–4751], 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 [Random Matrices Theory Appl. 9 (2020), p. 26] and Song et al. [J. Math. Anal. Appl. 502 (2021), p. 22].