Isotropic self-consistent equations for mean-field random matrices

TL;DR

This paper introduces a versatile method for deriving isotropic local laws for mean-field random matrices, applicable regardless of the matrix's expectation, and demonstrates its effectiveness on Wigner matrices with arbitrary expectation.

Contribution

The paper develops a simple, general approach for establishing isotropic local laws in mean-field random matrices, extending results to matrices with arbitrary expectation.

Findings

Established local laws for Wigner matrices with arbitrary expectation

Provided a probabilistic derivation of self-consistent equations

Method is insensitive to the matrix's expectation

Abstract

We present a simple and versatile method for deriving (an)isotropic local laws for general random matrices constructed from independent random variables. Our method is applicable to mean-field random matrices, where all independent variables have comparable variances. It is entirely insensitive to the expectation of the matrix. In this paper we focus on the probabilistic part of the proof -- the derivation of the self-consistent equations. As a concrete application, we settle in complete generality the local law for Wigner matrices with arbitrary expectation.