Multiple collisions of eigenvalues and singular values of matrix Gaussian field

Abstract

Let $X^β$ be a real symmetric or complex Hermitian matrix whose entries are independent Gaussian random fields. We provide the sufficient and necessary conditions such that multiple collisions of eigenvalue processes of $A^β+ T_βX^βT_β^*$ occur with positive probability. In addition, for a real or complex rectangular matrix $W^β$ with independent Gaussian random field entries, we obtain the sufficient and necessary conditions under which the probability of multiple collisions of non-trivial singular value processes of $B^β+ T_βW^β\tilde T_β$ is positive. In both cases, the size of the set of collision times is characterized via Hausdorff dimension.