Universality of local eigenvalue statistics in random matrices with external source

TL;DR

This paper proves that local eigenvalue statistics of certain random matrices with an external source follow universal patterns, despite the failure of global laws, using a four moment theorem approach.

Contribution

It establishes the universality of local eigenvalue statistics for a broad class of Wigner matrices with external sources, even when the global semicircle law does not hold.

Findings

Universal sine kernel formula for correlation functions

Local eigenvalue statistics are more resilient than global laws

Four moment theorem adapted for matrices with external sources

Abstract

Consider a random matrix of the form $W_n = M_n + D_n$, where $M_n$ is a Wigner matrix and $D_n$ is a real deterministic diagonal matrix ($D_n$ is commonly referred to as an external source in the mathematical physics literature). We study the universality of the local eigenvalue statistics of $W_n$ for a general class of Wigner matrices $M_n$ and diagonal matrices $D_n$. Unlike the setting of many recent results concerning universality, the global semicircle law fails for this model. However, we can still obtain the universal sine kernel formula for the correlation functions. This demonstrates the remarkable phenomenon that local laws are more resilient than global ones. The universality of the correlation functions follows from a four moment theorem, which we prove using a variant of the approach used earlier by Tao and Vu.