Random bistochastic matrices

Valerio Cappellini, Hans-Jürgen Sommers, Wojciech Bruzda, Karol Życzkowski
2009 Journal of Physics A: Mathematical and Theoretical  
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
more » ... 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.
doi:10.1088/1751-8113/42/36/365209 fatcat:wrjlot2nibh37mhjjitbxxwu2a