Combinatorics for general kinetically constrained spin models

TL;DR

This paper explores the configuration space of general kinetically constrained spin models, extending combinatorial analysis to higher dimensions and non-oriented models, and applies results to prove a conjecture in the field.

Contribution

It provides a new combinatorial framework for analyzing general KCMs, broadening understanding beyond the one-dimensional East model and addressing non-oriented dynamics.

Findings

Solved a generalized combinatorial problem for KCM configurations

Extended analysis to higher-dimensional, non-oriented models

Contributed to proving a significant conjecture in the field

Abstract

We study the set of possible configurations for a general kinetically constrained model (KCM), a non monotone version of the $\mathcal{U}$-bootstrap percolation cellular automata. We solve a combinatorial question that is a generalization of a problem addressed by Chung, Diaconis and Graham in 2001 for a specific one-dimensional KCM, the East model. Since the general models we consider are in any dimension and lack the oriented character of the East dynamics, we have to follow a completely different route than the one taken by Chung, Diaconis and Graham. Our combinatorial result is used by Mar\^ech\'e, Martinelli and Toninelli to complete the proof of a conjecture put forward by Morris.