Asymptotic Estimates for the Number of Contingency Tables, Integer Flows, and Volumes of Transportation Polytopes

We prove an asymptotic estimate for the number of m × n non-negative integer matrices (contingency tables) with prescribed row and column sums and, more generally, for the number of integer-feasible flows in a network. Similarly, we estimate the volume of the polytope of m × n non-negative real matrices with prescribed row and column sums. Our estimates are solutions of convex optimization problems, and hence can be computed efficiently. As a corollary, we show that if row sums R = (r1, …, rm) and column sums C = (c1, …, cn) with r1 + ⋯ + rm = c1 + ⋯ + cn = Nare sufficiently far from constant vectors, then, asymptotically, in the uniform probability space of the m × nnon-negative integer matrices with the total sum N of entries, the event consisting of the matrices with row sums R and the event consisting of the matrices with column sums C are positively correlated.