Random doubly stochastic matrices: The circular law

TL;DR

This paper proves that the eigenvalue distribution of normalized random doubly stochastic matrices converges to the circular law, confirming a longstanding conjecture in random matrix theory.

Contribution

It establishes the almost sure convergence of the spectral distribution of normalized doubly stochastic matrices to the circular law, confirming a conjecture by Chatterjee, Diaconis, and Sly.

Findings

Spectral distribution converges to the circular law.

Confirms the conjecture on eigenvalue distribution for doubly stochastic matrices.

Provides theoretical proof of spectral behavior in high dimensions.

Abstract

Let $X$ be a matrix sampled uniformly from the set of doubly stochastic matrices of size $n\times n$. We show that the empirical spectral distribution of the normalized matrix $\sqrt{n}(X-{\mathbf {E}}X)$ converges almost surely to the circular law. This confirms a conjecture of Chatterjee, Diaconis and Sly.