On a front evolution problem for the multidimensional East model

We consider a natural front evolution problem the East process on Z, 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 vmax(x), vmin(x) be the maximal/minimal velocities at which S(t) grows in that direction. If x is independent of q, we prove that vmax(x) = vmin(x) = γ(d)(1+o(1)) as q → 0,where γ(d) is the spectral gap of the process on Z. We also analyse the case in which some of the coordinates of x vanish as q → 0. In particular, for d = 2 we prove that if x approaches one of the two coordinate directions fast enough, then vmax(x) = vmin(x) = γ(1)(1+o(1)) = γ(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 Z+. A key ingredient of our analysis is the renormalisation technique of [11] 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.