Kinetically constrained spin models on trees

TL;DR

This paper investigates kinetically constrained spin models on trees, identifying ergodicity regimes, establishing critical densities via bootstrap percolation, and proving spectral gap positivity with novel martingale techniques, with implications for lattice models.

Contribution

It introduces a new martingale-based method to prove spectral gap positivity in kinetically constrained models, linking ergodicity to bootstrap percolation on trees.

Findings

Critical densities match bootstrap percolation thresholds.

Spectral gap is positive throughout the ergodic regime.

Method can be extended to lattice models.

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 $\mathbb{Z}^d$.