Cutoff at the "entropic time" for sparse Markov chains

TL;DR

This paper establishes the cutoff phenomenon for convergence to equilibrium in sparse Markov chains within random environments, revealing that mixing times depend on the number of states and the entropy of transition probabilities.

Contribution

It introduces a general framework for analyzing cutoff in non-reversible, sparse Markov chains with unknown equilibrium, including models with exchangeable entries and random environments.

Findings

Cutoff occurs at a time proportional to the logarithm of the number of states.

Mixing time is inversely related to the average row entropy of the transition matrix.

Results apply to models with i.i.d. transition rows in the domain of attraction of a Poisson-Dirichlet law.

Abstract

We study convergence to equilibrium for a large class of Markov chains in random environment. The chains are sparse in the sense that in every row of the transition matrix $P$ the mass is essentially concentrated on few entries. Moreover, the random environment is such that rows of $P$ are independent and such that the entries are exchangeable within each row. This includes various models of random walks on sparse random directed graphs. The models are generally non reversible and the equilibrium distribution is itself unknown. In this general setting we establish the cutoff phenomenon for the total variation distance to equilibrium, with mixing time given by the logarithm of the number of states times the inverse of the average row entropy of $P$. As an application, we consider the case where the rows of $P$ are i.i.d. random vectors in the domain of attraction of a Poisson-Dirichlet law with index $\alpha\in(0,1)$. Our main results are based on a detailed analysis of the weight of the trajectory followed by the walker. This approach offers an interpretation of cutoff as an instance of the concentration of measure phenomenon.