On a front evolution problem for the multidimensional East model

We consider a natural front evolution problem for the East process on Z(d), d >= 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)(x), v(min)(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)(x) = v(min)(x)((1+o(1))) = gamma((1+o(1)))(d) as q -> 0, where gamma(d) is the spectral gap of the process on Z(d). We also analyse the case in which x depends on q and some of its coordinates vanish as q -> 0. In particular, for d = 2 we prove that if x approaches one of the two coordinate directions fast enough, then v(max)(x) = v(min)(x)((1+o(1))) = gamma((1+o(1)))(1) = gamma(d(1+o(1)))(d) , 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 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 -> 0 of a suitable principal Dirichlet eigenvalue of the process.