Cutoff for the Averaging process on the hypercube and complete bipartite graphs

TL;DR

This paper investigates the cutoff phenomenon in the averaging process on hypercubes and complete bipartite graphs, revealing sharp convergence behaviors in these structures.

Contribution

It establishes cutoff results for the averaging process on hypercubes and bipartite graphs, extending understanding of mixing times in these graph classes.

Findings

Cutoff occurs in the averaging process on hypercubes.

Cutoff occurs in the averaging process on complete bipartite graphs.

Provides general insights into the averaging process on arbitrary graphs.

Abstract

We consider the averaging process on a graph, that is the evolution of a mass distribution undergoing repeated averages along the edges of the graph at the arrival times of independent Poisson processes. We establish cutoff phenomena for both the $L^1$ and $L^2$ distance from stationarity when the graph is a discrete hypercube and when the graph is complete bipartite. Some general facts about the averaging process on arbitrary graphs are also discussed.