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

• 成果类型: Article
• 署名作者: Cancrini, N.; Martinelli, F.; Roberto, C.; Toninelli, C.
• 署名单位: University of L'Aquila; Roma Tre University; Universite Paris Saclay; Centre National de la Recherche Scientifique (CNRS); Sorbonne Universite; Universite Paris Cite; Centre National de la Recherche Scientifique (CNRS)
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-014-0548-x
• 发表日期: 2015
• 页码: 247-266
• 关键词: bootstrap percolation

摘要

On the rooted -ary tree we consider a - 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 vacant. For this process the following picture was conjectured to hold. As long as is below the percolation threshold the process is ergodic with a finite relaxation time while, for , 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 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 behavior in the depth of the tree at and (ii) power law scaling in when approaches 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.