# Edge Universality for non-Hermitian Random Matrices

## TL;DR

This paper proves that the local eigenvalue statistics near the spectral edge of large non-Hermitian random matrices match those of the Gaussian Ginibre ensemble, establishing a universality result similar to the Tracy-Widom law for Hermitian matrices.

## Contribution

It demonstrates the universality of eigenvalue statistics at the spectral edge for non-Hermitian matrices with i.i.d. entries, extending known results to the non-Hermitian setting.

## Key findings

- Eigenvalue statistics near the spectral edge match Ginibre ensemble
- Universality holds for real and complex non-Hermitian matrices
- Results extend Tracy-Widom type universality to non-Hermitian matrices

## Abstract

We consider large non-Hermitian real or complex random matrices $X$ with independent, identically distributed centred entries. We prove that their local eigenvalue statistics near the spectral edge, the unit circle, coincide with those of the Ginibre ensemble, i.e. when the matrix elements of $X$ are Gaussian. This result is the non-Hermitian counterpart of the universality of the Tracy-Widom distribution at the spectral edges of the Wigner ensemble.