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arXiv · 2507.02298

Local laws and spectral properties of deformed sparse random matrices

Ji Oon Lee·Inyoung Yeo
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Abstract

We consider deformed sparse random matrices of the form $H= W+ λV$, where $W$ is a real symmetric sparse random matrix, $V$ is a random or deterministic, real, diagonal matrix whose entries are independent of $W$, and $λ= O(1) $ is a coupling constant. Under mild assumptions on the matrix entries of $W$ and $V$, we prove local laws for $H$ that compare the empirical spectral measure of it with a refined version of the deformed semicircle law. By applying the local laws, we also prove several spectral properties of $H$, including the rigidity of the eigenvalues and the asymptotic normality of the extremal eigenvalues.

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Ji Oon Lee, Inyoung Yeo. 2026-04-29. Local laws and spectral properties of deformed sparse random matrices. https://arxiv.org/abs/2507.02298

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