MIXING TIME OF A KINETICALLY CONSTRAINED SPIN MODEL ON TREES: POWER LAW SCALING AT CRITICALITY

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.
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Probability Theory and Related Fields, Springer Verlag, 2015
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N. Cancrini, F. Martinelli, C. Roberto, C. Toninelli. MIXING TIME OF A KINETICALLY CONSTRAINED SPIN MODEL ON TREES: POWER LAW SCALING AT CRITICALITY. Probability Theory and Related Fields, Springer Verlag, 2015. 〈hal-01696432〉

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