The asymptotic volume of the Birkhoff polytope

Let $m,ngeq 1$ be integers. Define $mathcal{T}_{m,n}$ to be the transportation polytope consisting of the $m imes n$ non-negative real matrices whose rows each sum to $1$ and whose columns each sum to $m/n$. The special case $mathcal{B}_n = mathcal{T}_{n,n}$ is the much-studied Birkhoff-von Neumann polytope of doubly-stochastic matrices. Using a recent asymptotic enumeration of non-negative integer matrices (Canfield and McKay, 2007), we determine the asymptotic volume of $mathcal{T}_{m,n}$ as $n oinfty$ with $m = m(n)$ such that $m/n$ neither decreases nor increases too quickly. In particular, we give an asymptotic formula for the volume of $mathcal{B}_n$.