Multiple collisions of eigenvalues and singular values of matrix Gaussian field

Abstract

Let $X^{\beta }$ 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 $A^{\beta }+T_{\beta }{X}^{\beta }{T}_{\beta }^{\ast }$ occur with positive probability. In addition, for a real or complex rectangular matrix $W^{\beta }$ 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 $B^{\beta }+T_{\beta }{W}^{\beta }{\tilde{T}}_{\beta }$ is positive. In both cases, the size of the set of collision times is characterized via Hausdorff dimension.

48 pages