Multiple collisions of eigenvalues and singular values of matrix Gaussian field

TL;DR

This paper investigates the conditions under which multiple eigenvalue and singular value collisions occur with positive probability in Gaussian matrix fields, characterizing the set of collision times via Hausdorff dimension.

Contribution

It provides necessary and sufficient conditions for multiple eigenvalue and singular value collisions in Gaussian matrix fields, including characterization of collision times.

Findings

Conditions for eigenvalue collisions are established.

Conditions for singular value collisions are established.

Hausdorff dimension characterizes collision time sets.

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 $A^\beta + T_\beta X^\beta T_\beta^*$ 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.