Linear time mixing and pre-cutoff for oriented kinetically constrained models

Abstract

We establish linear time precutoff for oriented kinetically constrained models (KCMs) on a $d$-dimensional box of side length $n$, whenever the finite-volume spectral gap is bounded away from zero uniformly in $n$. A typical example is the North-East model, a $0$-$1$ spin system on the two-dimensional integer lattice that evolves according to the following rule: whenever a site's southerly and westerly nearest neighbours have spin $0$, with rate one it resets its own spin by tossing a $(1-q)$-coin; at all other times its spin remains frozen. This settles, in the oriented case, a conjecture which states that there is linear time precutoff for all KCMs in the (open) ergodic parameter regime ($q>q_c$). The result has been shown recently, for general update families, in a perturbative regime ($q$ close to $1$). The previous general upper bound on the mixing time for oriented update families, in the full ergodic regime, was $O(n \log n)$. The proof combines a spectral estimate for a certain killed process with a graphical coupling to a stationary process and a backward witness path argument, exploiting the orientation of the constraints, to control the probability of failing to couple.