Optimal co-adapted coupling for the symmetric random walk on the hypercube

TL;DR

This paper identifies the fastest co-adapted coupling for two symmetric continuous-time random walks on an n-dimensional hypercube, providing insights into optimal coupling strategies in high-dimensional discrete spaces.

Contribution

It introduces and characterizes the optimal co-adapted coupling for symmetric random walks on the hypercube, advancing understanding of coupling efficiency in discrete structures.

Findings

The described coupling is proven to be the fastest within its class.

Provides a new method for analyzing coupling times in high-dimensional hypercubes.

Enhances theoretical understanding of stochastic processes on discrete graphs.

Abstract

Let X and Y be two simple symmetric continuous-time random walks on the vertices of the n-dimensional hypercube. We consider the class of co-adapted couplings of these processes, and describe an intuitive coupling which is shown to be the fastest in this class.