Empirical spectral distribution of a matrix under perturbation

TL;DR

This paper develops a perturbative expansion for the spectral distribution of large Hermitian matrices under small random perturbations, revealing different regimes linked to Gaussian free fields and free probability.

Contribution

It introduces a detailed perturbative framework for spectral distributions under perturbations, identifying multiple regimes and their connections to advanced probabilistic theories.

Findings

Different regimes depend on perturbation magnitude

Leading terms relate to Gaussian free field or free probability

Provides a unified perturbative approach for spectral analysis

Abstract

We provide a perturbative expansion for the empirical spectral distribution of a Hermitian matrix with large size perturbed by a random matrix with small operator norm whose entries in the eigenvector basis of the first one are independent with a variance profile. We prove that, depending on the order of magnitude of the perturbation, several regimes can appear, called perturbative and semi-perturbative regimes. Depending on the regime, the leading terms of the expansion are either related to the one-dimensional Gaussian free field or to free probability theory.