Random Bistochastic Matrices

TL;DR

This paper studies the properties and distributions of random bistochastic matrices, deriving explicit formulas for small cases, constructing ensembles for larger matrices, and estimating the volume of the Birkhoff polytope.

Contribution

It introduces methods to generate and analyze random bistochastic matrices, including explicit distributions for small matrices and an ensemble approach for larger ones.

Findings

Explicit formulas for N=2 case distributions

A new ensemble construction for larger matrices

An estimation of the Birkhoff polytope volume

Abstract

Ensembles of random stochastic and bistochastic matrices are investigated. While all columns of a random stochastic matrix can be chosen independently, the rows and columns of a bistochastic matrix have to be correlated. We evaluate the probability measure induced into the Birkhoff polytope of bistochastic matrices by applying the Sinkhorn algorithm to a given ensemble of random stochastic matrices. For matrices of order N=2 we derive explicit formulae for the probability distributions induced by random stochastic matrices with columns distributed according to the Dirichlet distribution. For arbitrary $N$ we construct an initial ensemble of stochastic matrices which allows one to generate random bistochastic matrices according to a distribution locally flat at the center of the Birkhoff polytope. The value of the probability density at this point enables us to obtain an estimation of the volume of the Birkhoff polytope, consistent with recent asymptotic results.