Gaussian perturbations of hard edge random matrix ensembles

Abstract

We study the eigenvalue correlations of random Hermitian matrices of the form , where H is a GUE matrix, , and M is a positive-definite Hermitian random matrix, independent of H, whose eigenvalue density is a polynomial ensemble. We show that there is a soft-to-hard edge transition in the microscopic behaviour of the eigenvalues of S close to 0 if ϵ tends to 0 together with at a critical speed, depending on the random matrix M. In a double scaling limit, we obtain a new family of limiting eigenvalue correlation kernels. We apply our general results to the cases where (i) M is a Laguerre/Wishart random matrix, (ii) M = G*G with G a product of Ginibre matrices, (iii) M = T *T with T a product of truncations of Haar distributed unitary matrices, and (iv) the eigenvalues of M follow a Muttalib–Borodin biorthogonal ensemble.