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Random Transpositions on Contingency Tables

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Abstract

The space of contingency tables with n total entries and fixed row and column sums is in bijection with parabolic double cosets of \(S_n\). Via this correspondence, the uniform distribution on \(S_n\) induces the Fisher–Yates distribution on contingency tables, which is classical for its use in the chi-squared test for independence. This paper studies the Markov chain on contingency tables induced by the random transpositions chain on \(S_n\). We show that the eigenfunctions are polynomials, which gives new insight into the orthogonal polynomials of the Fisher–Yates distribution. The eigenfunctions are used for upper bounds on the mixing time in special cases, as well as a general lower bound. The Markov chain on contingency tables is a novel example for the theory of double coset Markov chains.

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Acknowledgements

The author thanks Persi Diaconis for helpful discussion and suggestions and Zhihan Li for recognizing how contingency tables correspond to permutation matrices.

Funding

This work was partially supported by a National Defense Science & Engineering Graduate Fellowship and a Lieberman Fellowship at Stanford University.

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  1. Mackenzie Simper

    Present address: Washington University in St. Louis, St. Louis, MO, USA

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  1. Department of Mathematics, Stanford University, Stanford, CA, USA

    Mackenzie Simper

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Simper, M. Random Transpositions on Contingency Tables. J Theor Probab (2023). https://doi.org/10.1007/s10959-023-01286-1

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