Cutoff for the bidirectional East Process

TL;DR

This paper investigates the cutoff phenomenon for a hypercube-based process related to the East Process, establishing the cutoff time as proportional to the hypercube dimension with a specific window size.

Contribution

It introduces a new analysis of the cutoff for a hypercube process with Poisson clocks, extending understanding of mixing times in related models.

Findings

Cutoff occurs at time L/v where L is the hypercube dimension.

The cutoff window is of order √L.

Results are compared to the East Process and other non-local variants.

Abstract

This paper will examine the cutoff for a random process on the hypercube, $\{0, 1\}^L$, closely related to the East Process. In this process, every coordinate has two 1/2-Poisson clocks at each coordinate which add the coordinate to the previous or next one when they ring. We show that the cutoff is $L/v$ with a window of order $\sqrt{L}$, where $v$ is the speed of the front. We compare these results to the cutoff for the East Process and the cutoff for a non-local version of this same process studied by Ganguly, Lubetzky, and Martinelli as well as Ben-Hamou and Peres.