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**The power of averaging at two consecutive time steps: proof of a mixing conjecture by Aldous and Fill.**
*(English.
French summary)*
Zbl 1382.60095

Summary: Let \((X_{t})_{t=0}^{\infty}\) be an irreducible reversible discrete-time Markov chain on a finite state space \(\Omega\). Denote its transition matrix by \(P\). To avoid periodicity issues (and thus ensuring convergence to equilibrium) one often considers the continuous-time version of the chain \((X_{t}^{\mathrm{c}})_{t\geq 0}\) whose kernel is given by \(H_{t}:=e^{-t}\sum_{k}(tP)^{k}/k!\). Another possibility is to consider the associated averaged chain \((X_{t}^{\mathrm{ave}})_{t=0}^{\infty}\), whose distribution at time \(t\) is obtained by replacing \(P^{t}\) by \(A_{t}:=(P^{t}+P^{t+1})/2\).{

} A sequence of Markov chains is said to exhibit (total-variation) cutoff if the convergence to stationarity in total-variation distance is abrupt. Let \((X_{t}^{(n)})_{t=0}^{\infty}\) be a sequence of irreducible reversible discrete-time Markov chains. In this work we prove that the sequence of associated continuous-time chains exhibits total-variation cutoff around time \(t_{n}\) iff the sequence of the associated averaged chains exhibits total-variation cutoff around time \(t_{n}\). Moreover, we show that the width of the cutoff window for the sequence of associated averaged chains is at most that of the sequence of associated continuous-time chains. In fact, we establish more precise quantitative relations between the mixing-times of the continuous-time and the averaged versions of a reversible Markov chain, which provide an affirmative answer to a problem raised by D. Aldous and J. Fill [Reversible Markov Chains and Random Walks on Graphs. Berkeley: University of California (2002)].

} A sequence of Markov chains is said to exhibit (total-variation) cutoff if the convergence to stationarity in total-variation distance is abrupt. Let \((X_{t}^{(n)})_{t=0}^{\infty}\) be a sequence of irreducible reversible discrete-time Markov chains. In this work we prove that the sequence of associated continuous-time chains exhibits total-variation cutoff around time \(t_{n}\) iff the sequence of the associated averaged chains exhibits total-variation cutoff around time \(t_{n}\). Moreover, we show that the width of the cutoff window for the sequence of associated averaged chains is at most that of the sequence of associated continuous-time chains. In fact, we establish more precise quantitative relations between the mixing-times of the continuous-time and the averaged versions of a reversible Markov chain, which provide an affirmative answer to a problem raised by D. Aldous and J. Fill [Reversible Markov Chains and Random Walks on Graphs. Berkeley: University of California (2002)].

### MSC:

60J10 | Markov chains (discrete-time Markov processes on discrete state spaces) |

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\textit{J. Hermon} and \textit{Y. Peres}, Ann. Inst. Henri Poincaré, Probab. Stat. 53, No. 4, 2030--2042 (2017; Zbl 1382.60095)

### References:

[1] | D. Aldous and J. Fill.Reversible Markov Chains and Random Walks on Graphs. University of California, Berkeley, 2002. Available athttps://www.stat.berkeley.edu/ aldous/RWG/book.pdf. |

[2] | N. Alon and J. H. Spencer.The Probabilistic Method. John Wiley & Sons, New York, 2004. · Zbl 1333.05001 |

[3] | R. Basu, J. Hermon and Y. Peres. Characterization of cutoff for reversible Markov chains.Ann. Probab.45(3) (2017) 1448-1487. · Zbl 1374.60129 |

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[8] | E. Lubetzky and A. Sly. Cutoff phenomena for random walks on random regular graphs.Duke Math. J.153(3) (2010) 475-510. · Zbl 1202.60012 |

[9] | Y. Peres and P. Sousi. Mixing times are hitting times of large sets.J. Theoret. Probab.28(2) (2015) 488-519. · Zbl 1323.60094 |

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