The asymptotic volume of the Birkhoff polytope

Let m,n ≥ 1 be integers. Define Tm,n to be the transportation polytope consisting of the m × n non-negative real matrices whose rows each sum to 1 and whose columns each sum to m/n. The special case Bn = Tn,n is the much-studied Birkhoffvon 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 Tm,n as n → ∞ 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 Bn.