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On some mixing times for nonreversible finite Markov chains

  • Lu-Jing Huang (a1) and Yong-Hua Mao (a1)

Abstract

By adding a vorticity matrix to the reversible transition probability matrix, we show that the commute time and average hitting time are smaller than that of the original reversible one. In particular, we give an affirmative answer to a conjecture of Aldous and Fill (2002). Further quantitive properties are also studied for the nonreversible finite Markov chains.

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Corresponding author

* Postal address: Laboratory of Mathematics and Complex Systems, Ministry of Education, School of Mathematical Sciences, Beijing Normal University, Beijing 100875, P. R. China.
** Email address: maoyh@bnu.edu.cn

References

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[1] Aldous, D. J. and Fill, J. A. (2002). Reversible Markov Chains and Random Walks on Graphs. Preprint. Available at https://www.stat.berkeley.edu/~aldous/RWG/book.html.
[2] Avrachenkov, K., Cottatellucci, L., Maggi, L. and Mao, Y.-H. (2013). Maximum entropy mixing time of circulant Markov processes. Statist. Prob. Lett. 83, 768773.
[3] Bierkens, J. (2016). Non-reversible Metropolis-Hastings. Statist. Comput. 26, 12131228.
[4] Chen, T.-L. and Hwang, C.-R. (2013). Accelerating reversible Markov chains. Statist. Prob. Lett. 83, 19561962.
[5] Cui, H. and Mao, Y.-H. (2010). Eigentime identity for asymmetric finite Markov chains. Front. Math. China 5, 623634.
[6] Diaconis, P., Holmes, S. and Neal, R. M. (2000). Analysis of a nonreversible Markov chain sampler. Ann. Appl. Prob. 10, 726752.
[7] Gaudillière, A. and Landim, C. (2014). A Dirichlet principle for non reversible Markov chains and some recurrence theorems. Prob. Theory Relat. Fields 158, 5589.
[8] Hwang, C.-R., Hwang-Ma, S.-Y. and Sheu, S. J. (1993). Accelerating Gaussian diffusions. Ann. Appl. Prob. 3, 897913.
[9] Hwang, C.-R., Hwang-Ma, S.-Y. and Sheu, S.-J. (2005). Accelerating diffusions. Ann. Appl. Prob. 15, 14331444.

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On some mixing times for nonreversible finite Markov chains

  • Lu-Jing Huang (a1) and Yong-Hua Mao (a1)

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