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The Random \((n-k)\)-Cycle to Transpositions Walk on the Symmetric Group

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Abstract

We study the rate of convergence of the Markov chain on \(S_n\) which starts with a random \((n-k)\)-cycle for a fixed \(k \ge 1\), followed by random transpositions. The convergence to the stationary distribution turns out to be of order n. We show that after \(cn + \frac{\ln k}{2}n\) steps for \(c>0\), the law of the Markov chain is close to the uniform distribution. The character of the defining representation is used as test function to obtain a lower bound for the total variation distance. We identify the asymptotic distribution of the test function given the law of the Markov chain for the \((n-1)\)-cycle case. The upper bound relies on estimates for the difference of normalized characters.

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Acknowledgements

The author would like to thank Jason Fulman for suggesting the problem and his most valuable comments.

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  1. Department of Mathematics, University of Southern California, Los Angeles, CA, 90089-2532, USA

    Alperen Y. Özdemir

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  1. Alperen Y. Özdemir
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Correspondence to Alperen Y. Özdemir.

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Özdemir, A.Y. The Random \((n-k)\)-Cycle to Transpositions Walk on the Symmetric Group. J Theor Probab 32, 1438–1460 (2019). https://doi.org/10.1007/s10959-018-0826-0

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  • DOI: https://doi.org/10.1007/s10959-018-0826-0

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