On the number of contingency tables and the independence heuristic

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