Multiple collisions of eigenvalues and singular values of matrix Gaussian field

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_β X^β T_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_β W^β T_β is positive. In both cases, the size of the set of collision times is characterized via Hausdorff dimension.