On the number of contingency tables and the independence heuristic

Abstract

We obtain sharp asymptotic estimates on the number of $n\times n$ contingency tables with two linear margins $Cn$ and $BCn$. The results imply a second order phase transition on the number of such contingency tables, with a critical value at \ts $B_c:=1+\sqrt{1+1/C}$. As a consequence, for \ts $B>B_c$, we prove that the classical \emph{independence heuristic} leads to a large undercounting.

10 pages, 1 figure