Random Bistochastic Matrices

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.

💡 Research Summary

The paper investigates ensembles of random stochastic matrices and their doubly‑stochastic (bistochastic) counterparts, focusing on the probability measure induced on the Birkhoff polytope—the convex set of all N × N doubly‑stochastic matrices. A stochastic matrix has non‑negative entries whose columns each sum to one; its columns can be drawn independently from any distribution (the authors mainly consider Dirichlet distributions). By contrast, a doubly‑stochastic matrix must satisfy both row‑ and column‑sum constraints, which introduces correlations among the columns. To generate random bistochastic matrices from a given ensemble of stochastic matrices, the authors employ the Sinkhorn–Knopp algorithm (also known simply as the Sinkhorn algorithm). This iterative procedure alternately normalises rows and columns of a non‑negative matrix; under mild conditions it converges to a unique bistochastic matrix. The mapping from the original stochastic ensemble to the Birkhoff polytope thus defines a new probability measure on the polytope.

For the simplest non‑trivial case N = 2, the authors derive closed‑form expressions for the induced distribution. Assuming each column of the original stochastic matrix follows a Dirichlet(α) law, the Sinkhorn transformation reduces to a simple ratio adjustment, and the resulting bistochastic matrix entries follow a Beta distribution. The authors compute the exact density at the centre of the polytope (the matrix with all entries equal to ½) and describe the full shape of the distribution, providing a benchmark for numerical simulations.

For arbitrary dimension N, the authors construct a specific initial ensemble of stochastic matrices that yields a “locally flat” distribution near the centre of the Birkhoff polytope. By choosing the Dirichlet parameters appropriately (essentially α = 1, i.e., a uniform distribution over the simplex for each column), the Sinkhorn‑generated bistochastic matrices are almost uniformly distributed in a neighbourhood of the centre. This property enables the evaluation of the probability density at the centre point. Since the total probability integrates to one, the density at the centre multiplied by the volume of the polytope gives an estimate of that volume. Using this approach, the authors obtain a numerical estimate of the Birkhoff polytope volume that agrees with recent asymptotic formulas derived by other methods. In particular, their estimate matches the known leading‑order term \