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, $μ_n$. The sequence is said to exhibit $μ_n$-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 $μ_n$-cutoff for an arbitrary sequence of initial distributions $μ_n$ (in the above setup). Our characterization is expressed in terms of hitting times of sets which are "worst" w.r.t. $μ_n$. Consider a Markov chain on $Ω$ whose stationary distribution in $π$. Let $t_{\mathrm{H}}(α) :=\max_{x \in Ω,A \subset Ω:\,π(A) \ge α}\mathbb{E}_{x}[T_{A}]$ be the expected hitting time of the worst set of size at least $α$. 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 $μ_n$-cutoff can be characterized in terms of concentration of hitting times (starting from $μ_n$) of sets which are worst in expectation w.r.t. $μ_n$. 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 $ε( t_{\mathrm{H}}(ε)-t_{\mathrm{H}}(1-ε)) \le Ct_{\mathrm{rel}} |\log ε|$, for all $0< ε< 1/2$, where $t_{\mathrm{rel}} $ is the inverse of the spectral gap of the chain.