The Brown measure of the sum of a self-adjoint element and an elliptic element

Abstract

We completely determine the Brown measure of the sum of a self-adjoint element and an elliptic element, which is the limiting eigenvalue distribution of the random matrix \[Y_N+\sqrt{s-\frac{t}{2}}X_N+i\sqrt{\frac{t}{2}}X_N'\] where $Y_N$ is an $N\times N$ deterministic Hermitian matrix whose eigenvalue distribution converges as $N\to \infty$ and $X_N$ and $X_N^{\prime }$ are independent Gaussian unitary ensembles. We also study various asymptotic behaviors of this Brown measure as the variance of the elliptic element approaches infinity.

References in corpus (3)

• Local limit theorems in free probability theory
• The Brown measure of the sum of a self-adjoint element and an imaginary multiple of a semicircular element
• A Novel Approach to Non-Hermitian Random Matrix Models

Cited by in corpus (2)

• Brown measure of the sum of an elliptic operator and a free random variable in a finite von Neumann algebra
• The Brown measure of unbounded variables with free semicircular imaginary part