Phase transition in random contingency tables with non-uniform margins

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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.