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A technical report on hitting times, mixing and cutoff. (English) Zbl 1388.60137
Summary: Consider a sequence of continuous-time irreducible reversible Markov chains and a sequence of initial distributions, $$\mu_n$$. Instead of performing a worst case analysis, one can study the rate of convergence to the stationary distribution starting from these initial distributions. The sequence is said to exhibit (total variation) $$\mu_n$$-cutoff if the convergence to stationarity in total variation distance is abrupt, w.r.t. this sequence of initial distributions.
In this work we give a characterization of $$\mu_n$$-cutoff (and also of total-variation mixing) for an arbitrary sequence of initial distributions $$\mu_n$$ (in the above setup). Our characterization is expressed in terms of hitting times of sets which are “worst” (in some sense) w.r.t. $$\mu_n$$.
Consider a Markov chain on $$\Omega$$ whose stationary distribution is $$\pi$$. Let $$t_{\mathrm{H}}(\alpha) :=\max_{x\in\Omega, A\subseteq \Omega:\pi(A)\geqslant\alpha}\mathbb E_x[T_A]$$ be the expected hitting time of the set of stationary probability at least $$\alpha$$ which is “worst in expectation” (starting from the worst starting state). The connection between $$t_{\mathrm{H}}(\cdot)$$ and the mixing time of the chain was previously studied by D. J. Aldous [J. Lond. Math. Soc., II. Ser. 25, 564–576 (1982; Zbl 0489.60077)] and later by L. Lovász and P. Winkler [DIMACS, Ser. Discrete Math. Theor. Comput. Sci. 41, 85–133 (1998; Zbl 0908.60065)], and was recently refined by Y. Peres and P. Sousi [J. Theor. Probab. 28, No. 2, 488–519 (2015; Zbl 1323.60094)] and independently by R. I. Oliveira [Electron. J. Probab. 17, Paper No. 70, 12 p. (2012; Zbl 1251.60059)]. In this work we further refine this connection and show that $$\mu_n$$-cutoff can be characterized in terms of concentration of hitting times (starting from $$\mu_n$$) of sets which are worst in expectation w.r.t. $$\mu_n$$. Conversely, we construct a counter-example which demonstrates that in general cutoff (as opposed to cutoff w.r.t. a certain sequence of initial distributions) cannot be characterized in this manner.
Finally, we also prove that there exists an absolute constant $$C$$ such that for any Markov chain $$_\varepsilon(t_{\mathrm{H}}(\varepsilon) - t_{\mathrm{H}}(1 -\varepsilon))\leqslant C t_{\mathrm{rel}}|\log \varepsilon|$$, for all $$0 < \varepsilon < 1/2$$, where $$t_{\mathrm{rel}}$$ is the inverse of the spectral gap of the additive symmetrization $$\frac{1}{2}(P + P^\ast)$$.

##### MSC:
 60J27 Continuous-time Markov processes on discrete state spaces 60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
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##### References:
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