Phase transition in random contingency tables with non-uniform margins
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For parameters n,δ,B, and C, let X=(X_kℓ) be the random uniform contingency table whose first n^δ rows and columns have margin BCn and the last n rows and columns have margin Cn . For every 0<δ<1, we establish a sharp phase transition of the limiting distribution of each entry of X at the critical value B_c=1+√(1+1/C). In particular, for 1/2<δ<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 E[X_11] is uniformly bounded for B<B_c, but has sharp asymptotic C(B-B_c) n^1-δ for B>B_c. We also establish a strong law of large numbers for the row sums in top right and top left blocks.
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