On a front evolution problem for the multidimensional East model

Abstract

We consider a natural front evolution problem for 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 x, let $v_{\max }(\mathbf{x}),v_{\min }(\mathbf{x})$ be the maximal/minimal velocities at which $S(t)$ grows in that direction. If x is independent of q, we prove that $v_{\max }(\mathbf{x})=v_{\min }{(\mathbf{x})}^{(1+o(1))}={\mathit{\gamma }}_d^{(1+o(1))}$ as $q\to 0$, where ${\mathit{\gamma }}_d$ is the spectral gap of the process on ${\mathbb{Z}}^d$. We also analyse the case in which x depends on q and some of its coordinates vanish as $q\to 0$. In particular, for $d=2$ we prove that if x approaches one of the two coordinate directions fast enough, then $v_{\max }(\mathbf{x})=v_{\min }{(\mathbf{x})}^{(1+o(1))}={\mathit{\gamma }}_1^{(1+o(1))}={\mathit{\gamma }}_d^{d(1+o(1))}$, i.e. the growth of $S(t)$ close to the coordinate directions is much slower than the growth in the bulk and it 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 [12] to estimate the spectral gap of the East process. A main novelty here is the extension of this technique to get the main asymptotic as $q\to 0$ of a suitable principal Dirichlet eigenvalue of the process.