Bulk Universality for Real Matrices with Independent and Identically Distributed Entries

Abstract

We consider real, Gauss-divisible matrices $A_t=A+\sqrt{t}B$, where $B$ is from the real Ginibre ensemble. We prove that the bulk correlation functions converge to a universal limit for $t=O(N^{-1/3+\epsilon })$ if $A$ satisfies certain local laws. If $A=\frac{1}{\sqrt{N}}({\xi }_{jk}{)}_{j,k=1}^N$ with ${\xi }_{jk}$ independent and identically distributed real random variables having zero mean, unit variance and finite moments, the Gaussian component can be removed using local laws proven by Bourgade--Yau--Yin, Alt--Erdős--Krüger and Cipolloni--Erdős--Schröder and the four moment theorem of Tao--Vu.

Revised version