Front evolution of the Fredrickson-Andersen one spin facilitated model

Abstract

The Fredrickson-Andersen one spin facilitated model (FA-1f) on Z belongs to the class of kinetically constrained spin models (KCM). Each site refreshes with rate one its occupation variable to empty (respectively occupied) with probability q (respectively p = 1 − q), provided at least one nearest neighbor is empty. Here, we study the non equilibrium dynamics of FA-1f started from a configuration entirely occupied on the left half-line and focus on the evolution of the front, namely the position of the leftmost zero. We prove, for q larger than a thresholdq < 1, a law of large numbers and a central limit theorem for the front, as well as the convergence to an invariant measure of the law of the process seen from the front.Let Ω = {0, 1} Z be the space of configurations andbe the subspace of configurations with a leftmost zero at the origin. For a configuration σ for which there exists x ∈ Z such that for every y < x, σ(y) = 1 and σ(x) = 0, we call x the front of configuration σ and we denote it by X(σ). For Λ ⊂ Z and σ ∈ Ω, we denote by σ Λ the restriction of σ to the set Λ. For σ ∈ Ω and x ∈ Z, let σ x be the configuration σ

Some recent advances for limit theorems

“…ϕ(e −t y) ϕ(e −(t+s) y) = ϕ t+s (y), and µ is solution of µ = T t µ * µ t . The convolution identity (14) implies that the family of operators (P t ) t≥0 defined, for all h ∈ C b (R) (the space of bounded continuous functions on R), by…”

“…In [14], FA-1f model is considered on Z starting from a configuration which has an empty site at the origin and is completely occupied in the left half line and the evolution of the front, namely the position of the leftmost empty site, is investigated. The idea explored in [14] is to couple FA-1f dynamics with a contact process (where, in this setting, empty sites correspond to infected sites) in order to use the well-known behavior of the contact process (ballistic motion of the front, shape theorem for the coupled zone) to study the FA-1f dynamics. Due to the fact that there is no constraint for the contact process for the move empty to occupied but there is one such constraint for the FA-1f, we can couple FA-1f and contact trajectories in such a way that FA-1f configurations contain more empty sites than contact process configurations.…”

“…Finally in Section 3, we deal with another discrete model presenting the propagation of an infection on a discrete graph; in this case, we are interested in the set of infected vertices and the limit theorems describe the asymptotic shape of such a set. In particular, we deal with first passage percolation model from [32] in Section 3.1, with the contact process from [48] in Section 3.2 and with non-attractive model such as Fredrickson-Andersen one spin facilitated model in dimension one from [14] in Section 3.3. [115,116], the method relies on the corresponding Stein equation in function f :…”

Dynamics of interacting particle systems

“…In this section, we intend to present a progress in the understanding of kinetically constrained models (KCM), based on a joint work with Oriane Blondel and Cristina Toninelli [7]. We focus on the Fredrickson-Andersen one spin facilitated model (FA-1f ) where the constraint requires at least one empty nearest neighbor.…”

Exponential convergence to equilibrium for the $d$-dimensional East model

“…Our results, which are the first to hold for a KCM in dimension greater than 1 and for any p, may help to understand further the out-of-equilibrium behavior of the d-dimensional East model. Indeed, such an exponential relaxation result was key to proving "shape theorems" in one-dimensional models in [2,10,4].…”