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

  • Alperen Y. Özdemir
Article

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.

Keywords

Markov chain Convergence rate Symmetric group Defining representation Asymptotic distribution Murnaghan–Nakayama Rule 

Mathematics Subject Classification (2010)

60C05 

Notes

Acknowledgements

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

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Southern CaliforniaLos AngelesUSA

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