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 crit- icality. 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.

Cancrini, N., Martinelli, F., Roberto, C., & Toninelli, C. (2015). Mixing time of a kinetically constrained spin model on trees: power law scaling at criticality. PROBABILITY THEORY AND RELATED FIELDS, 161(1-2), 247-266 [10.1007/s00440-014-0548-x].

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

MARTINELLI, Fabio;
2015

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 crit- icality. 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.
Cancrini, N., Martinelli, F., Roberto, C., & Toninelli, C. (2015). Mixing time of a kinetically constrained spin model on trees: power law scaling at criticality. PROBABILITY THEORY AND RELATED FIELDS, 161(1-2), 247-266 [10.1007/s00440-014-0548-x].
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11590/136792
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