What Does a Random Contingency Table Look Like?

Let R = (r1, . . . , rm) and C = (c1, . . . , cn) be positive integer vectors such that r1 + . . .+ rm = c1 + . . .+ cn. We consider the set Σ(R, C) of non-negative m × n integer matrices (contingency tables) with row sums R and column sums C as a finite probability space with the uniform measure. We prove that a random table D ∈ Σ(R, C) is close with high probability to a particular matrix (“typical table”) Z defined as follows. We let g(x) = (x + 1) ln(x + 1) − x lnx for x ≥ 0 and let g(X) = P ij g(xij) for a non-negative matrix X = (xij). Then g(X) is strictly concave and attains its maximum on the polytope of non-negative m × n matrices X with row sums R and column sums C at a unique point, which we call the typical table Z.