O ct 2 01 0 PROPERTIES OF UNIFORM DOUBLY STOCHASTIC MATRICES

We investigate the properties of uniform doubly stochastic random matrices, that is non-negative matrices conditioned to have their rows and columns sum to 1. The rescaled marginal distributions are shown to converge to exponential distributions and indeed even large sub-matrices of side-length o(n) behave like independent exponentials. We determine the limiting empirical distribution of the singular values the the matrix. Finally the mixing time of the associated Markov chains is shown to be exactly 2 with high probability. Random matrices have become a central area of focus for modern probability theory and numerous models have been intensely studied including Wigner, Wishart, GOE and GUE matrices [3]. In this paper we study a model for which much less is known, namely uniformly chosen entries of the set of doubly stochastic matrices (called Uniformly Distributed Stochastic Matrices). The Birkhoff polytope is an (n− 1)2 dimensional polytope in Rn2 constituting the set of doubly stochastic matrices and is the convex hull of the permutation matrices (see e.g. [41]). While its extreme points are sparse matrices we shall see that typical entries chosen according to the uniform distribution are by contrast very dense. Little is known about the properties of uniformly distributed stochastic matrices as they fall outside the scope of techniques from the usual random matrix theory, however, important recent progress has been made by Barvinok and Hartigan. We will let X = (Xij)i,j=1,...,n denote a uniform doubly stochastic matrix. By symmetry its rows and columns are exchangeable and all its entries have the same marginal distribution. It is natural then to ask what is the limiting distribution of nX11, the first entry rescaled to have mean 1. In our first result we determine that the rescaled marginal distribution converges to an exponential random variable of mean 1. Theorem 1. With X = (Xij)i,j=1,...,n a uniformly chosen doubly stochastic matrix we have that, nX11 d → exp(1) as n → ∞ where the convergence is in total variation distance. Further, for any ǫ > 0, dtv(nX11, exp(1)) = O(n −1/2+ǫ). A natural extension to this question is to ask about the joint distribution for a collection of several entries. It can be shown using the same approach that finite collections of random variables converge to independent exponentials with mean 1. This convergence holds not just in distribution but also in total-variation distance and its moments converge to the moments of independent exponentials (see Section 3.2). We believe that in many ways uniformly distributed stochastic