Gaussian perturbations of hard edge random matrix ensembles

Abstract

We study the eigenvalue correlations of random Hermitian $n\times n$ matrices of the form $S=M+\epsilon H$, where $H$ is a GUE matrix, $\epsilon >0$, 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 $\epsilon$ tends to $0$ together with $n\to +\infty$ 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) $M=G^{\ast }G$ with $G$ a product of Ginibre matrices, (iii) $M=T^{\ast }T$ with $T$ a product of truncations of Haar distributed unitary matrices, and (iv) the eigenvalues of $M$ follow a Muttalib-Borodin biorthogonal ensemble.

36 pages, 8 figures

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Cited by in corpus (2)

• Additive Matrix Convolutions of Pólya Ensembles and Polynomial Ensembles
• Singular values for products of two coupled random matrices: hard edge phase transition