Multiple collisions of eigenvalues and singular values of matrix Gaussian field

Let Xβ be a real symmetric or complex Hermitian matrix whose entries are independent anisotropic Gaussian random fields. We provide the sufficient and necessary conditions such that multiple collisions of eigenvalue processes of Aβ+TβXβTβ∗ occur with positive probability. In addition, for a real or complex rectangular matrix Wβ 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 Bβ+TβWβT˜β is positive. In both cases, the size of the set of collision times is characterized via Hausdorff dimension. Moreover, we recover the corresponding results where the Gaussian random fields are isotropic.