Kinetically constrained spin models

Abstract

We analyze the density and size dependence of the relaxation time for kinetically constrained spin models (KCSM) intensively studied in the physics literature as simple models sharing some of the features of the glass transition. KCSM are interacting particle systems on Z d with Glauber-like dynamics, reversible w.r.t. a simple product i.i.d Bernoulli(p) measure. The essential feature of a KCSM is that the creation/destruction of a particle at a given site can occur only if the current configuration around it satisfies certain constraints which completely define each specific model. No other interaction is present in the model. From the mathematical point of view, the basic issues concerning positivity of the spectral gap inside the ergodicity region and its scaling with the particle density p remained open for most KCSM (with the notably exception of the East model in d = 1; Aldous and Diaconis in J Stat Phys 107(5-6):945-975, 2002). Here for the first time we: (i) identify the ergodicity region by establishing a connection with an associated bootstrap percolation model; 460 N. Cancrini et al.(ii) develop a novel multi-scale approach which proves positivity of the spectral gap in the whole ergodic region; (iii) establish, sometimes optimal, bounds on the behavior of the spectral gap near the boundary of the ergodicity region and (iv) establish pure exponential decay at equilibrium for the persistence function, i.e. the probability that the occupation variable at the origin does not change before time t. Our techniques are flexible enough to allow a variety of constraints and our findings disprove certain conjectures which appeared in the physical literature on the basis of numerical simulations.

Facilitated Oriented Spin Models: Some Non Equilibrium Results

“…As a consequence the dynamics becomes slower at higher density and an ergodicity breaking transition may occur at a finite critical density, p c < 1. This threshold, as it has been formalized in Section 2.3 of [3], corresponds to the lowest density at which the origin belongs with finite probability to a cluster of particles which are mutually and forever blocked due to the constraints. Among the above models the North-East is the only one displaying such a transition at p c < 1 (see [13] and [3]).…”

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

“…As shown in [16], our approach has a good chance to apply also to other models with an ergodicity phase transition, notably the North-East model on Z 2 for which the critical density p c coincides with the oriented percolation threshold [3]. .…”

“…As soon as k ≥ 2, the model shares some of the key features of another well known kinetically constrained system, namely the North East model [3,13]. More specifically, since above the critical density p c = 1/k the occupied vertices begin to percolate (under the reversible measure µ), blocked clusters appear and time ergodicity is lost.…”

“…This model belongs to the class of kinetically constrained spin models which have been introduced in physics literature to model liquid/glass transition or, more generally, glassy dynamics (see [10,17] for physical background and [3] for related mathematical work). As for most of the kinetically constrained models, the Bernoulli(p) product measure µ is a reversible measure for the process.…”

Dynamic heat capacity of the east model and of a bead-spring polymer model.

“…5. where 1/(2ln(2)) ≤ B ≤ 1/ln(2) [23,24]. Since log(τ) ~ 1/T 2 ≠ 1/T, an Angell plot for the east model would have characteristically fragile curvature.…”