Mixing time bounds for oriented kinetically constrained spin models

TL;DR

This paper establishes that oriented kinetically constrained spin models, like the North-East model, have mixing times of order n log n under certain conditions, shedding light on their dynamic behavior relevant to glass physics.

Contribution

It provides the first rigorous proof of mixing time bounds for a class of oriented kinetically constrained spin models, confirming conjectures about their dynamics.

Findings

Mixing time is O(n log n) when relaxation time is O(1).

Results support the shape conjecture for these models.

Models exhibit glass-like dynamic features.

Abstract

We analyze the mixing time of a class of oriented kinetically constrained spin models (KCMs) on a d-dimensional lattice of $n^d$ sites. 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 p-coin, at all other times its spin remains frozen. Such models are very popular in statistical physics because, in spite of their simplicity, they display some of the key features of the dynamics of real glasses. We prove that the mixing time is O(n log n) whenever the relaxation time is O(1). Our study was motivated by the "shape" conjecture put forward by G. Kordzakhia and S.P. Lalley.