Universality of extremal eigenvalues of large random matrices

TL;DR

This paper proves that the largest eigenvalue's spectral radius of large random matrices with i.i.d. entries universally follows the Gumbel distribution, regardless of the entry distribution, confirming a long-standing conjecture.

Contribution

It establishes the first universality result for extremal eigenvalues in random matrix theory, including the Gumbel law for spectral radius and Poisson process for extremal eigenvalues.

Findings

Spectral radius follows Gumbel law universally

Argument of the largest eigenvalue is uniform on the unit circle

Extremal eigenvalues form a Poisson point process

Abstract

We prove that the spectral radius of a large random matrix $X$ with independent, identically distributed complex entries follows the Gumbel law irrespective of the distribution of the matrix elements. This solves a long-standing conjecture of Bordenave and Chafa{\"\i} and it establishes the first universality result for one of the most prominent extremal spectral statistics in random matrix theory. Furthermore, we also prove that the argument of the largest eigenvalue is uniform on the unit circle and that the extremal eigenvalues of $X$ form a Poisson point process.