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 $t=O(N^{-1∕3+\mathit{ϵ}})$ if A satisfies certain local laws. If $A=\frac{1}{\sqrt{N}}{({\mathrm{\xi }}_{jk})}_{j,k=1}^N$ with ${\mathrm{\xi }}_{jk}$ 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.