A technical report on hitting times, mixing and cutoff

Abstract

Consider a sequence of continuous-time irreducible reversible Markov chains and a sequence of initial distributions, $\mu_n$. The sequence is said to exhibit $\mu_n$-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 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" w.r.t. $\mu_n$. Consider a Markov chain on $\Omega$ whose stationary distribution in $\pi$. Let $t_{\mathrm{H}}(\alpha) :=\max_{x \in \Omega,A \subset \Omega :\,\pi(A) \ge \alpha}\mathbb{E}_{x}[T_{A}]$ be the expected hitting time of the worst set of size at least $\alpha$. It was recently proved by Peres and Sousi and independently by Oliveira that $t_{\mathrm{H}}(1/4) $ captures the order of the mixing time. 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 every Markov chain $\epsilon( t_{\mathrm{H}}(\epsilon)-t_{\mathrm{H}}(1-\epsilon)) \le Ct_{\mathrm{rel}} |\log \epsilon|$, for all $0< \epsilon < 1/2$, where $t_{\mathrm{rel}} $ is the inverse of the spectral gap of the chain.