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Transactions of the American Mathematical Society

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Phase transition in random contingency tables with non-uniform margins


Authors: Samuel Dittmer, Hanbaek Lyu and Igor Pak
Journal: Trans. Amer. Math. Soc. 373 (2020), 8313-8338
MSC (2010): Primary 54C40, 14E20; Secondary 46E25, 20C20
DOI: https://doi.org/10.1090/tran/8094
Published electronically: October 5, 2020
MathSciNet review: 4177260
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Abstract: 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.


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Additional Information

Samuel Dittmer
Affiliation: Department of Mathematics, University of California Los Angeles, Los Angeles, California 90095
MR Author ID: 1057891
Email: samuel.dittmer@math.ucla.edu

Hanbaek Lyu
Affiliation: Department of Mathematics, University of California Los Angeles, Los Angeles, California 90095
Email: hlyu@math.ucla.edu

Igor Pak
Affiliation: Department of Mathematics, University of California Los Angeles, Los Angeles, California 90095
MR Author ID: 293184
ORCID: 0000-0001-8579-7239
Email: pak@math.ucla.edu

Keywords: Differential geometry, algebraic geometry
Received by editor(s): April 11, 2019
Received by editor(s) in revised form: July 16, 2019, and August 15, 2019
Published electronically: October 5, 2020
Additional Notes: The third author was partially supported by the NSF
Article copyright: © Copyright 2020 American Mathematical Society