On Properties of Random Binary Contingency Tables with Non-Uniform Margin

We study the random binary contingency tables with non-uniform margin. More precisely, for parameters $n,\delta ,B,C$, we consider $X=(X_{ij})$ with $X_{ij}\in \{0,1\}$, the random binary contingency tables whose first $[n^{\delta }]$ rows and columns have margin $[BCn]$ and the rest columns and rows have margin $[Cn]$. We study various asymptotic properties of $X$ as $n\to \mathrm{\infty }$. This answers a question posted by Barvinok.