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October 2013 Kinetically constrained spin models on trees
F. Martinelli, C. Toninelli
Ann. Appl. Probab. 23(5): 1967-1987 (October 2013). DOI: 10.1214/12-AAP891


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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F. Martinelli. C. Toninelli. "Kinetically constrained spin models on trees." Ann. Appl. Probab. 23 (5) 1967 - 1987, October 2013.


Published: October 2013
First available in Project Euclid: 28 August 2013

zbMATH: 1284.60170
MathSciNet: MR3134727
Digital Object Identifier: 10.1214/12-AAP891

Primary: 60K35, 82C20

Rights: Copyright © 2013 Institute of Mathematical Statistics


Vol.23 • No. 5 • October 2013
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