We introduce the kinetically constrained models (KCM), a class of interacting particle systems with a simple spin flip dynamics subject to local dynamical constraints. Each vertex is resampled (independently) at rate one by tossing a (1-q)-coin iff a certain neighbourhood of the vertex contains no particles. In other words, the holes (empty vertices) act as facilitating sites. When q shrinks to 0, the presence of the constraints gives rise to glassy dynamics, in particular to an anomalous divergence of the characteristic time scales. Thus, KCM are extensively used in physics literature to model the liquid-glass transition, a longstanding open problem in condensed matter physics.

We focus on the behavior of E(T_0), the mean over the stationary process of the first time at which the origin becomes empty. Our results establish the universality classes of KCM in two dimensions: we group all possible constraints into distinct classes with all models in a class featuring the same divergence for E(T_0) as q->0. Within each class, we present an efficient relaxation mechanism that involves the cooperative motion of large rare patches of empty sites and we use it to determine matching upper and lower bounds for E(T_0).

Joint work with L.Marêché, I.Hartarsky, F.Martinelli and R.Morris

The Fredrickson-Andersen 2-spin facilitated model (FA-2f) on the d-dimensional lattice, d ≥ 2, is a paradigmatic interacting particle system with kinetic constraints (KCM) featuring "dynamical facilitation", an important mechanism in condensed matter physics. In FA-2f a site may change its state only if at least two of its nearest neighbors are empty. The process, while reversible w.r.t. a product Bernoulli measure, has degenerate jumps rates and it is non-attractive, with an anomalous divergence of characteristic time scales as the density q of the empty sites tends to zero. A natural random variable encoding the above features is the first time T at which the origin becomes empty for the stationary process. Our main result is the sharp threshold T ≍ Exp[ d λ(d,2)/q^(1/(d-1)) ] w.h.p. as q → 0 , with λ(d,2) the threshold constant for the 2-neighbor bootstrap percolation, the monotone deterministic cellular automaton counterpart of FA-2f. This is the first sharp result for a critical KCM. Besides settling various controversies accumulated in the physics literature over the last four decades, our novel techniques enable completing the recent ambitious program on the universality phenomenon for critical KCM and establishing sharp thresholds for other two-dimensional KCM.

Joint work with I. Hartarski and C. Toninelli