Mixing time bounds for oriented kinetically constrained spin models

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.

Mixing times for a constrained Ising process on the two-dimensional torus at low density

“…The kinetically constrained Ising process (KCIP) refers to a class of interacting particle systems introduced by physicists in [12,13] to study the glass transition. These processes have also appeared outside of the computer science literature (see the surveys [5,6] for examples). In this paper, we analyze one of the simplest and most-studied processes introduced in [12,13], called the FA1f process.…”

“…KCIP models have attracted a great deal of interest recently, including applications to combinatorics, computer science, and other areas. The recent survey [14] discusses KCIPs throughout physics, while [6,5] have useful surveys of places that the KCIP has appeared outside of the physics literature. Recent mathematical progress has included new bounds on the mixing properties of the KCIP in various regimes [19,2,22,6,3,5,4], and the very recent work [23] makes substantial progress towards a "universal" approach for bounding relaxation times of kinetically-constrained processes.…”

Mixing times for a constrained Ising process on the torus at low density

“…We contrast our work with specific papers. Many previous results, such as [CM13,CFM15], deal primarily with the regime in which p is a constant, independently of n. In this regime, many KCIPs mix relatively quickly and the GLM], study the small-p regime, but only in one dimension. In particular, the methods employed in [AD02] completely break down for d > 1 and thus are not applicable to our setting.…”

“…Versions of this process have accrued other names since then, including the kinetically constrained spin model, the east model [AD02] and the north-east model [Val04]. These models have attracted a great deal of interest recently, including applications to combinatorics, computer science, and other areas; [CM13,CFM14a] have useful surveys of places that the KCIP has appeared outside of the physics literature. Recent mathematical progress has included new bounds on the mixing time of the KCIP in various different regimes [KL06, CMRT09, MT13, CM13, CFM14b, CFM14a, CFM15].…”

“…We contrast our work with specific papers. Many previous results, such as [CM13,CFM15], deal primarily with the regime in which p is a constant, independently of n. In this regime, many KCIPs mix relatively quickly and the…”