Local laws and spectral properties of deformed sparse random matrices

TL;DR

The work analyzes deformed sparse random matrices $H=W+\lambda V$ and proves local laws that compare the empirical spectrum to a refined deformed semicircle law, capturing edge behavior. By developing a cumulant-expansion framework and introducing a refined measure via free convolution, the authors obtain averaged and entrywise local laws up to the spectral edge, along with eigenvalue rigidity and precise edge fluctuations. They show that for a broad sparsity range, the extremal eigenvalues concentrate near refined classical locations and, when $V$ is random and $\lambda$ is sufficiently large, exhibit asymptotically Gaussian fluctuations with a variance linked to the base measure $\nu$. The results extend the understanding of spectral statistics in sparse, deformed random matrices and provide sharp descriptions of edge behavior and finite-size corrections, with potential implications for related random-graph and quantum-chaos models.

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.

Theorems & Definitions (86)

• Remark 1.1: Notational conventions
• Definition 1.2: High probability event
• Definition 1.3: Stochastic domination
• Definition 2.1: Sparse random matrices
• Remark 2.2
• Remark 2.6
• Definition 2.7
• Lemma 2.8
• Lemma 2.9
• Proposition 2.10: Theorem 4.1, BB