Mixing time of a kinetically constrained spin model on trees: power law scaling at criticality

TL;DR

This paper rigorously analyzes the critical behavior of a kinetically constrained spin model on trees, revealing power law scaling of relaxation times at and near the critical point, advancing understanding of critical dynamics in such models.

Contribution

It provides the first rigorous analysis of a kinetically constrained model's behavior at criticality, showing power law scaling of relaxation times.

Findings

Power law growth of relaxation time at criticality.

Power law scaling in $(p_c-p)^{-1}$ near the critical point.

Close parallels with spin glass models at criticality.

Abstract

On the rooted $k$-ary tree we consider a 0-1 kinetically constrained spin model in which the occupancy variable at each node is re-sampled with rate one from the Bernoulli(p) measure iff all its children are empty. For this process the following picture was conjectured to hold. As long as $p$ is below the percolation threshold $p_c=1/k$ the process is ergodic with a finite relaxation time while, for $p>p_c$, the process on the infinite tree is no longer ergodic and the relaxation time on a finite regular sub-tree becomes exponentially large in the depth of the tree. At the critical point $p=p_c$ the process on the infinite tree is still ergodic but with an infinite relaxation time. Moreover, on finite sub-trees, the relaxation time grows polynomially in the depth of the tree. The conjecture was recently proved by the second and forth author except at criticality. Here we analyse the critical and quasi-critical case and prove for the relevant time scales: (i) power law behaviour in the depth of the tree at $p=p_c$ and (ii) power law scaling in $(p_c-p)^{-1}$ when $p$ approaches $p_c$ from below. Our results, which are very close to those obtained recently for the Ising model at the spin glass critical point, represent the first rigorous analysis of a kinetically constrained model at criticality.