## Combinatorics, Probability and Computing

Cambridge Journals Online - CUP Full-Text Page
Combinatorics, Probability and Computing (2010), 19:517-539 Cambridge University Press
doi:10.1017/S0963548310000039

Paper

## What Does a Random Contingency Table Look Like?

 ALEXANDER BARVINOKa1†a1 Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1043, USA (e-mail: barvinok@umich.edu)
 Article author query barvinok a [Google Scholar]

## Abstract

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 ln x for x ≥ 0 and let g(X) = ∑i,jg(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.