A sequence of Markov chains is said to exhibit (total variation) cutoff if the convergence to stationarity in total variation distance is abrupt. We consider reversible lazy chains. We prove a necessary and sufficient condition for the occurrence of the cutoff phenomena in terms of concentration of hitting time of “worst” (in some sense) sets of stationary measure at least αα, for some α∈(0,1)α∈(0,1). We also give general bounds on the total variation distance of a reversible chain at time tt in terms of the probability that some “worst” set of stationary measure at least αα was not hit by time tt. As an application of our techniques, we show that a sequence of lazy Markov chains on finite trees exhibits a cutoff iff the product of their spectral gaps and their (lazy) mixing-times tends to ∞∞.
This aforementioned connection was substantially refined recently by Peres and Sousi ( @cite_1 Theorem 1.1) and independently by Oliveira ( @cite_5 Theorem 2). Their approach relied on the theory of random times to stationarity combined with a certain de-randomization" argument which shows that for any lazy reversible irreducible finite chain and any stopping time @math such that @math , @math . As a (somewhat indirect) consequence, they showed that for any @math (this was extended to @math in @cite_19 ), there exist some constants @math such that for any lazy reversible irreducible finite chain
@cite_19 @cite_5 @cite_1
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Consider a sequence of continuous-time irreducible reversible Markov chains and a sequence of initial distributions, @math . The sequence is said to exhibit @math -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 @math -cutoff for an arbitrary sequence of initial distributions @math (in the above setup). Our characterization is expressed in terms of hitting times of sets which are "worst" w.r.t. @math . Consider a Markov chain on @math whose stationary distribution in @math . Let @math be the expected hitting time of the worst set of size at least @math . It was recently proved by Peres and Sousi and independently by Oliveira that @math captures the order of the mixing time. In this work we further refine this connection and show that @math -cutoff can be characterized in terms of concentration of hitting times (starting from @math ) of sets which are worst in expectation w.r.t. @math . 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 @math such that for every Markov chain @math , for all @math , where @math is the inverse of the spectral gap of the chain. Let @math . We show that that the mixing time of a continuous-time Markov chain on a finite state space is about as large as the largest expected hitting time of a subset of the state space with stationary measure @math . Suitably modified results hold in discrete time and or without the reversibility assumption. The key technical tool in the proof is the construction of random set @math such that the hitting time of @math is a light-tailed stationary time for the chain. We note that essentially the same results were obtained independently by Peres and Sousi. Given an irreducible discrete time Markov chain on a finite state space, we consider the largest expected hitting time T(α) of a set of stationary measure at least α for α ∈ (0, 1). We obtain tight inequalities among the values of T(α) for different choices of α. One consequence is that T(α) ≤ T(1 2) α for all α < 1 2. As a corollary we have that if the chain is lazy in a certain sense as well as reversible, then T(1 2) is equivalent to the chain’s mixing time, answering a question of Peres. We furthermore demonstrate that the inequalities we establish give an almost everywhere pointwise limiting characterisation of possible hitting time functions T(α) over the domain α ∈ (0, 1 2]. 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.