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

2022/02/08 by Cheuk Yin Lee, Jian Song, Lee, Cheuk Yin +5
Mathematics · #FOS: Mathematics #Probability (math.PR) #math.PR

paper · pdf · doi:10.48550/arxiv.2202.03625

25 pages

arxiv created 2022/02/08 · arxiv updated 2022/02/09

Abstract

Let X= \X(t), t ∈ \mathbb R^N\ be a centered Gaussian random field with values in \mathbb R^d satisfying certain conditions and let F ⊂ \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 ( X(t) ∈ F \hbox for some t ∈ \mathbb R^N ) = 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].

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