CHARACTERIZATION OF CUTOFF FOR REVERSIBLE MARKOV CHAINS

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 $\alpha$, for some $\alpha \in (0,1)$. We also give general bounds on the total variation distance of a reversible chain at time $t$ in terms of the probability that some "worst" set of stationary measure at least $\alpha$ was not hit by time $t$. As an application of our techniques we show that a sequence of lazy Markov chains on finite trees exhibits a cutoff iff the ratio of their relaxation-times and their (lazy) mixing-times tends to 0

Cutoff for non-backtracking random walks on sparse random graphs

A finite ergodic Markov chain is said to exhibit cutoff if its distance to stationarity remains close to 1 over a certain number of iterations and then abruptly drops to near 0 on a much shorter time scale. Discovered in the context of card shuffling (Aldous-Diaconis, 1986), this phenomenon is now believed to be rather typical among fast mixing Markov chains. Yet, establishing it rigorously often requires a challengingly detailed understanding of the underlying chain. Here we consider non-backtracking random walks on random graphs with a given degree sequence. Under a general sparsity condition, we establish the cutoff phenomenon, determine its precise window, and prove that the (suitably rescaled) cutoff profile approaches a remarkably simple, universal shape

Cutoff stability of multivariate geometric Brownian motion

This article quantifies the asymptotic $\varepsilon$-mixing times, as $\varepsilon$ tends to 0, of a multivariate stable geometric Brownian motion with respect to the Wasserstein-Kantorovich-Rubinstein-2-distance. We study the cases of commutative drift and diffusion coeffcient matrices.Comment: 9 page

The power of averaging at two consecutive time steps: Proof of a mixing conjecture by Aldous and Fill

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 $(X_t^{\mathrm{c}})_{t \ge 0}$ 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 $P$ 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 Aldous and Fill.Non

A Spectral Characterization for Concentration of the Cover Time

Abstract: We prove that for a sequence of finite vertex-transitive graphs of increasing sizes, the cover times are asymptotically concentrated if and only if the product of the spectral gap and the expected cover time diverges. In fact, we prove this for general reversible Markov chains under the much weaker assumption (than transitivity) that the maximal hitting time of a state is of the same order as the average hitting time