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

Ching-Wei Ho

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arXiv · 2007.06100

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

Ching-Wei Ho

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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 $X_N'$ are independent Gaussian unitary ensembles. We also study various asymptotic behaviors of this Brown measure as the variance of the elliptic element approaches infinity.

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BibTeX RIS

Ching-Wei Ho. 2020-07-12. The Brown measure of the sum of a self-adjoint element and an elliptic element. https://arxiv.org/abs/2007.06100

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