# What does a random contingency table look like?

Research paper by **Alexander Barvinok**

Indexed on: **25 Nov '09**Published on: **25 Nov '09**Published in: **Mathematics - Combinatorics**

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#### Abstract

Let R=(r_1, ..., r_m) and C=(c_1, ..., c_n) be positive integer vectors such
that r_1 +... + r_m=c_1 +... + c_n. We consider the set Sigma(R, C) of
non-negative mxn 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 in Sigma(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 non-negative x and let g(X)=sum_ij g(x_ij) for a
non-negative matrix X=(x_ij). Then g(X) is strictly concave and attains its
maximum on the polytope of non-negative mxn matrices X with row sums R and
column sums C at a unique point, which we call the typical table Z.