Random walk on sparse random digraphs

TL;DR

This paper investigates the cutoff phenomenon for random walks on sparse directed graphs, establishing precise conditions, the cutoff window, and the universal shape of the cutoff profile, along with describing the equilibrium measure.

Contribution

It rigorously proves the cutoff phenomenon for non-reversible random walks on sparse directed graphs under the configuration model, including the cutoff window and profile.

Findings

Cutoff phenomenon is established for these random walks.

The cutoff window and profile are precisely characterized.

A detailed description of the equilibrium measure is provided.

Abstract

A finite ergodic Markov chain exhibits cutoff if its distance to equilibrium remains close to its initial value over a certain number of iterations and then abruptly drops to near 0 on a much shorter time scale. Originally discovered in the context of card shuffling (Aldous-Diaconis, 1986), this remarkable phenomenon is now rigorously established for many reversible chains. Here we consider the non-reversible case of random walks on sparse directed graphs, for which even the equilibrium measure is far from being understood. We work under the configuration model, allowing both the in-degrees and the out-degrees to be freely specified. We establish the cutoff phenomenon, determine its precise window and prove that the cutoff profile approaches a universal shape. We also provide a detailed description of the equilibrium measure.