Abstract
We consider irreducible reversible discrete time Markov chains on a finite state space. Mixing times and hitting times are fundamental parameters of the chain. We relate them by showing that the mixing time of the lazy chain is equivalent to the maximum over initial states \(x\) and large sets \(A\) of the hitting time of \(A\) starting from \(x\). We also prove that the first time when averaging over two consecutive time steps is close to stationarity is equivalent to the mixing time of the lazy version of the chain.
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Acknowledgments
We are indebted to Oded Schramm for suggesting the use of the parameter \(t_{\mathrm{G}}\) to relate mixing times and hitting times. We are grateful to David Aldous for helpful discussions. After this work was completed, we were informed that Theorem 1.1 was also proved independently by Roberto Imbuzeiro Oliveira ([6]).We thank Yang Cai, Júlia Komjáthy and the referee for useful comments.
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Peres, Y., Sousi, P. Mixing Times are Hitting Times of Large Sets. J Theor Probab 28, 488–519 (2015). https://doi.org/10.1007/s10959-013-0497-9
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DOI: https://doi.org/10.1007/s10959-013-0497-9