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arXiv · 2112.11382

Functional CLT for non-Hermitian random matrices

László Erdős·Hong Chang Ji
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Abstract

For large dimensional non-Hermitian random matrices $X$ with real or complex independent, identically distributed, centered entries, we consider the fluctuations of $f(X)$ as a matrix where $f$ is an analytic function around the spectrum of $X$. We prove that for a generic bounded square matrix $A$, the quantity $\mathrm{Tr}f(X)A$ exhibits Gaussian fluctuations as the matrix size grows to infinity, which consists of two independent modes corresponding to the tracial and traceless parts of $A$. We find a new formula for the variance of the traceless part that involves the Frobenius norm of $A$ and the $L^{2}$-norm of $f$ on the boundary of the limiting spectrum.

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László Erdős, Hong Chang Ji. 2021-12-21. Functional CLT for non-Hermitian random matrices. https://arxiv.org/abs/2112.11382

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