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--Erd\H{o}s--Kr\"{u}ger and Cipolloni--Erd\H{o}s--Schr\"{o}der and the four moment theorem of Tao--Vu.