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

Abstract

ABSTRACT. On the rooted k-ary tree we consider a 0-1 kinetically constrained spin model in which the occupancy variable at each node is re-sampled with rate one from the Bernoulli(p) measure iff all its children are empty. For this process the following picture was conjectured to hold. As long as p is below the percolation threshold pc = 1/k the process is ergodic with a finite relaxation time while, for p > pc, the process on the infinite tree is no longer ergodic and the relaxation time on a finite regular sub-tree becomes exponentially large in the depth of the tree. At the critical point p = pc the process on the infinite tree is still ergodic but with an infinite relaxation time. Moreover, on finite sub-trees, the relaxation time grows polynomially in the depth of the tree.The conjecture was recently proved by the second and forth author except at criticality. Here we analyse the critical and quasi-critical case and prove for the relevant time scales: (i) power law behaviour in the depth of the tree at p = pc and (ii) power law scaling in (pc − p) −1 when p approaches pc from below. Our results, which are very close to those obtained recently for the Ising model at the spin glass critical point, represent the first rigorous analysis of a kinetically constrained model at criticality.

Cutoff for the East Process

“…Previous work. Before proving our results we recall the main findings of [23] and [12]. We now formally define the critical density for the OFA-jf model:…”

Towards a universality picture for the relaxation to equilibrium of kinetically constrained models

“…Although this bound has been greatly improved for special choices of the update family U , yielding in some cases the correct behavior (cf. [10,13,14]), for general KCM and contrary to the situation of bootstrap percolation in two dimensions, there is yet no universality picture for the scaling of E µ (τ 0 ). The present paper represents the first step of a general project concerning KCM with update family U satisfying q c (U ) = 0, with the aim of establishing universality results on the scaling of E µ (τ 0 ) as q → 0 analogous to those proved within bootstrap percolation.…”

“…Constrained Poincaré inequalities for KCM, implying a positive spectral gap and exponential mixing, have already been established [12], mainly using the so-called halving method. Here, inspired by our previous analysis of KCM on trees [10,24], we develop an alternative method which, besides being more natural and direct, applies as well to update families with a large (depending on q) or infinite number of elements. As an example, in section 2.2 we prove a Poincaré inequality for the KCM for which the constraint requires that the oriented neighbours of the to-be-updated vertex belong to an infinite cluster of infected vertices.…”

Asymptotically Exact Solution of the Fredrickson-Andersen Model

“…Asymptotic Relaxation.-The asymptotic relaxation to zero, φ(t → ∞) → 0, on times t/t σ 1 is governed by a scaling function, φ(t → ∞) = φ(t/t σ ) [61]. For the ofa it has been shown that φ(t) ∝ exp(−t/τ α ) where τ α ∼ σ −γ and the exponent γ ≥ 2 could only be bounded from below [63]. In terms of the scaling function we find τ α ∼ t σ and in particular the preceding analysis determines the exponent γ ≡ γ compatible with the bound.…”