On a front evolution problem for the multidimensional East model

TL;DR

This paper studies the front evolution in the multidimensional East model, revealing how the growth velocities depend on the spectral gap and how the shape becomes elongated near coordinate directions, with implications for mixing times.

Contribution

It extends renormalisation techniques to analyze the spectral gap and asymptotics of the East process in multiple dimensions, providing new insights into front growth and shape.

Findings

Growth velocities relate to the spectral gap as q→0.

Near coordinate directions, growth is governed by the 1D process.

Established mixing time cutoff for finite boxes.

Abstract

We consider a natural front evolution problem the East process on $\mathbb{Z}^d, d\ge 2,$ a well studied kinetically constrained model for which the facilitation mechanism is oriented along the coordinate directions, as the equilibrium density $q$ of the facilitating vertices vanishes. Starting with a unique unconstrained vertex at the origin, let $S(t)$ consist of those vertices which became unconstrained within time $t$ and, for an arbitrary positive direction $\mathbf x,$ let $v_{\max}(\mathbf x),v_{\min}(\mathbf x )$ be the maximal/minimal velocities at which $S(t)$ grows in that direction. If $\mathbf x$ is independent of $q$, we prove that $v_{\max}(\mathbf x)= v_{\min}(\mathbf x)^{(1+o(1))}=\gamma(d) ^{(1+o(1))}$ as $q\to 0$, where $\gamma(d)$ is the spectral gap of the process on $\mathbb{Z}^d$. We also analyse the case in which some of the coordinates of $\mathbf x$ vanish as $q\to 0$. In particular, for $d=2$ we prove that if $\mathbf x$ approaches one of the two coordinate directions fast enough, then $v_{\max}(\mathbf x)= v_{\min}(\mathbf x)^{(1+o(1))}=\gamma(1) ^{(1+o(1))}=\gamma(d)^{d(1+o(1))},$ i.e. the growth of $S(t)$ close to the coordinate directions is dictated by the one dimensional process. As a result the region $S(t)$ becomes extremely elongated inside $\mathbb{Z}^d_+$. We also establish mixing time cutoff for the chain in finite boxes with minimal boundary conditions. A key ingredient of our analysis is the renormalisation technique of arXiv:1404.7257 to estimate the spectral gap of the East process. Here we extend this technique to get the main asymptotics of a suitable principal Dirichlet eigenvalue of the process.