Cutoff for East models

We consider the East model in $\mathbb Z^d$, an example of a kinetically constrained interacting particle system with oriented constraints, together with one of its natural variant. Under any ergodic boundary condition it is known that the mixing time of the chain in a box of side $L$ is $\Theta(L)$ for any $d\ge 1$. Moreover, with minimal boundary conditions and at low temperature, i.e. low equilibrium density of the facilitating vertices, the chain exhibits cutoff around the mixing time of the $d=1$ case. Here we extend this result to high equilibrium density of the facilitating vertices. As in the low density case, the key tool is to prove that the speed of infection propagation in the $(1,1,\dots,1)$ direction is larger than $d$ $\times$ the same speed along a coordinate direction. By borrowing a technique from first passage percolation, the proof links the result to the precise value of the critical probability of oriented (bond or site) percolation in $\mathbb Z^d$.