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. $P(X(t) \in F for some t \in R^N) = 0$ , 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].This is a joint work with Cheuk Yin Lee, Jian Song and Yimin Xiao.