Kinetically constrained spin models on trees

Abstract

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 ${\mathbb{Z}}^{d}$.