On a front evolution problem for the multidimensional East model

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.