A Short Note on Asymptotic Enumeration of Contingency Tables with Non-Uniform Margins

TL;DR

This paper derives precise asymptotic formulas for counting contingency tables with non-uniform margins, focusing on specific parameter regimes and expanding determinant calculations in the context of maximal entropy methods.

Contribution

It provides a detailed expansion of the determinant of quadratic forms in asymptotic formulas for contingency tables with non-uniform margins, advancing maximal entropy techniques.

Findings

Asymptotic enumeration formulas for contingency tables with specified margins.

Explicit determinant expansions in the asymptotic analysis.

Conditions under which the formulas are valid, notably when B < B_c.

Abstract

In this short note, we compute the precise asymptotics for the number of contingency tables with non-uniform margins. More precisely, for parameter $n,\delta, B,C>0$, we consider the set of matrices whose first $[n^\delta]$ rows and columns have sum $[BCn]$ and the rest $n$ rows and columns have sum $[Cn]$. We compute the precise asymptotics of the cardinality of this set when $B<B_c=1+\sqrt{1+1/C}$ using the maximal entropy methods developed by Barvinok and Hartigan. The only contribution of this note is a detailed expansion of the determinant of quadratic forms in asymptotic formulas.