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

Let X = {X(t), t is an element of RN} be a centered Gaussian random field with values in Rd satisfying certain conditions and let F subset of Rd be a Borel set. In our main theorem, we provide a sufficient condition for F to be polar for X, i.e. P(X(t) is an element of F for some t is an element of RN) = 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].