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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Phase transition in random contingency tables with non-uniform margins
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by Samuel Dittmer, Hanbaek Lyu and Igor Pak PDF
Trans. Amer. Math. Soc. 373 (2020), 8313-8338 Request permission

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
  • 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
  • © Copyright 2020 American Mathematical Society
  • 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
  • MathSciNet review: 4177260