Wangjun Yuan: Hitting probability of Gaussian random fields and collision of eigenvalues of random matrices

Abstract

Let \(X = \{ X(t),t \in R^N\}\) be a centered Gaussian random field with values in R^d satisfying certain conditions and let \(F \subset R^d\) be a Borel set. In the talk, we provide a sufficient condition for F to be polar for X, i.e., \(P(X(t) \in F \text{ for some } t \in R^N) = 0\) , which improves significantly the existence results. We provide a variety of examples of Gaussian random field for which our result is applicable. Moreover, 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].