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

Abstract

We study a kinetically constrained Ising process (KCIP) associated with a graph G and density parameter p; this process is an interacting particle system with state space {0, 1} G . The stationary distribution of the KCIP Markov chain is the Binomial(|G|, p) distribution on the number of particles, conditioned on having at least one particle. The 'constraint' in the name of the process refers to the rule that a vertex cannot change its state unless it has at least one neighbour in state '1'. The KCIP has been proposed by statistical physicists as a model for the glass transition, and more recently as a simple algorithm for data storage in computer networks. In this note, we study the mixing time of this process on the torus G = Z d L , d ≥ 3, in the low-density regime p = c |G| for arbitrary 0 < c < ∞; this regime is the subject of a conjecture of Aldous and is natural in the context of computer networks. Our results provide a counterexample to Aldous' conjecture, suggest a natural modification of the conjecture, and show that this modification is correct up to logarithmic factors. The methods developed in this paper also provide a strategy for tackling Aldous' conjecture for other graphs. ‡

Universality Results for Kinetically Constrained Spin Models in Two Dimensions

“…Another well-studied KCM, introduced by Friedrickson and Andersen [2], is the kfacilitated model (FA-kf), whose update family consists of the k-sets of nearest neighbours of the origin: a site can be updated iff it has at least k empty nearest neighbours. In this case it was proved in [20,33] that q c (Z d , U ) = 0 for all 1 k d, whereas q c (Z d , U ) = 1 for all k > d. Moreover, the relaxation time T rel (q) diverges as 1/q Θ (1) when k = 1 [10,35], and as a (k − 1)-times iterated exponential of q −1/(d−k+1) when 2 k d [27]. The above scalings also hold for the mean infection time E µ (τ 0 ).…”

A Note on the Spectral Gap of the Fredrickson–Andersen One Spin Facilitated Model

“…We study the Fredrickson–Andersen one spin facilitated model (FA1f), on the lattice following [ 1 ] and on finite graphs following [ 3 , 5 , 6 ]. The reader is referred to [ 1 , 3 , 5 , 6 ] for the relevant background, references, and complete introduction of the model.…”

“…A particularly interesting case, studied in [ 5 , 6 ] and more recently in [ 3 ], is when for some positive constant c , where V ( G ) denotes the set of sites in G . A lower bound on the spectral gap is given in [ 5 , 6 ] and later on in [ 3 ] by suggesting a relaxation mechanism in which vacancies travel as random walkers on G . The next theorem bounds the spectral gap of this model from above, showing that this mechanism has a leading contribution.…”

Cutoff for the Square Plaquette Model on a Critical Length Scale

“…Simulations and heuristic analysis based on this observation suggest that for the SPM the relaxation time scales like e cβ (Arrhenius scaling), and that the dynamics are closely related to those of the Fredrickson-Andersen KCM. The mixing properties of this model have been well studied (see [23,24,2] and references therein). On the other hand, in a related model called the triangular plaquette model the relaxation time is expected to scale like e cβ 2 (super-Arrhenius scaling) [11].…”