Title: Phase transition in random contingency tables with non-uniform margins
Speaker: Hanbaek Lyu (UCLA)
Abstract: Contingency tables are matrices with nonnegative integer entries with fixed row and column margins. Understanding the structure of uniformly chosen contingency table with given margins is an important problem especially in statistics. 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 other $n$ rows and columns have margin $\lfloor Cn \rfloor$. For any $0<\delta<1$, we establish a sharp phase transition of the limiting distribution of each entries of $X$ at the critical value $B_{c}=1+\sqrt{1+1/C}$. One of our main result shows that, for $1/2<\delta<1$, all entries have uniformly bounded expectation for $B<B_{c}$, but the mass concentrates at the smallest block and grows in the order of $n^{1-\delta}$ for $B>B_{c}$. We also establish a strong law of large numbers for the row sums within blocks.
Joint work with Igor Pak and Sam Dittmer.
