Phase transition in random contingency tables with non-uniform margins

For parameters $n,\delta,B,$ and $C$, let $X=(X_{k\ell})$ be the random uniform contingency table whose first $\lfloor n^{\delta} \rfloor $ rows and columns have margin $\lfloor BCn \rfloor$ and the last $n$ rows and columns have margin $\lfloor Cn \rfloor$. For every $0<\delta<1$, we establish a sharp phase transition of the limiting distribution of each entry of $X$ at the critical value $B_{c}=1+\sqrt{1+1/C}$. In particular, for $1/2<\delta<1$, we show that the distribution of each entry converges to a geometric distribution in total variation distance, whose mean depends sensitively on whether $B<B_{c}$ or $B>B_{c}$. Our main result shows that $\mathbb{E}[X_{11}]$ is uniformly bounded for $B<B_{c}$, but has sharp asymptotic $C(B-B_{c}) n^{1-\delta}$ for $B>B_{c}$. We also establish a strong law of large numbers for the row sums in top right and top left blocks.