Tracy-Widom limit for free sum of random matrices

TL;DR

This paper proves that the largest eigenvalues of a certain class of random matrices converge to the Tracy-Widom distribution, using Dyson Brownian motion and local laws, under mild assumptions on the matrices.

Contribution

It establishes the Tracy-Widom limit for the largest eigenvalue of free sums of random matrices with minimal regularity assumptions.

Findings

Largest eigenvalues follow Tracy-Widom distribution

Established optimal local law for Dyson Brownian motion

Provided a comparison method for Green functions

Abstract

We consider fluctuations of the largest eigenvalues of the random matrix model $A+UBU^{*}$ where $A$ and $B$ are $N \times N$ deterministic Hermitian (or symmetric) matrices and $U$ is a Haar-distributed unitary (or orthogonal) matrix. We prove that the largest eigenvalue weakly converges to the Tracy-Widom distribution, under mild assumptions on $A$ and $B$ to guarantee that the density of states of the model decays as square root around the upper edge. Our proof is based on the comparison of the Green function along the Dyson Brownian motion starting from the matrix $A + UBU^{*}$ and ending at time $N^{-1/3+\chi}$. As a byproduct of our proof, we also prove an optimal local law for the Dyson Brownian motion up to the constant time scale.