Random doubly stochastic matrices: The circular law

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.Comment: Published in at http://dx.doi.org/10.1214/13-AOP877 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org