On the number of contingency tables and the independence heuristic

Abstract

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