Abstract

Abstract. 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 positive integer vectors R = (r1,..., rm) and C = (c1,..., cn) with r1+...+rm = c1+...+cn = N are sufficiently generic, then in the uniform probability space of the m × n nonnegative integer matrices with the total sum N of entries, the event consisting of the matrices with the row sums R and the event consisting of the matrices with the column sums C attract exponentially in mn. Our main tools are a new estimate for the volume of a section of a simplex by a subspace of a small codimension and an integral representation for the number of contingency tables. 1. Introduction and

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