The Annals of Applied Probability

Kinetically constrained spin models on trees

F. Martinelli and C. Toninelli

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We analyze kinetically constrained $0$–$1$ spin models (KCSM) on rooted and unrooted trees of finite connectivity. We focus in particular on the class of Friedrickson–Andersen models FA-jf and on an oriented version of them. These tree models are particularly relevant in physics literature since some of them undergo an ergodicity breaking transition with the mixed first-second order character of the glass transition. Here we first identify the ergodicity regime and prove that the critical density for FA-jf and OFA-jf models coincide with that of a suitable bootstrap percolation model. Next we prove for the first time positivity of the spectral gap in the whole ergodic regime via a novel argument based on martingales ideas. Finally, we discuss how this new technique can be generalized to analyze KCSM on the regular lattice ${\mathbb{Z}}^{d}$.

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Ann. Appl. Probab., Volume 23, Number 5 (2013), 1967-1987.

First available in Project Euclid: 28 August 2013

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Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82C20: Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs

Kinetically constrained models dynamical phase transitions glass transition bootstrap percolation stochastic models on trees interacting particle systems


Martinelli, F.; Toninelli, C. Kinetically constrained spin models on trees. Ann. Appl. Probab. 23 (2013), no. 5, 1967--1987. doi:10.1214/12-AAP891.

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