Bootstrap percolation and kinetically constrained models on hyperbolic lattices

TL;DR

This paper investigates bootstrap percolation on hyperbolic lattices, revealing a nontrivial critical density and a mixed transition character, which differs from Euclidean lattice behavior and relates to glass and jamming transitions.

Contribution

It demonstrates the existence of a nontrivial bootstrap percolation transition on hyperbolic lattices and characterizes its mixed nature, extending understanding beyond Euclidean cases.

Findings

Existence of a nontrivial critical density $0< ho_c<1$ on hyperbolic lattices.

Transition is discontinuous but has diverging correlation length.

Behavior differs significantly from Euclidean lattices with trivial critical densities.

Abstract

We study bootstrap percolation (BP) on hyperbolic lattices obtained by regular tilings of the hyperbolic plane. Our work is motivated by the connection between the BP transition and the dynamical transition of kinetically constrained models, which are in turn relevant for the study of glass and jamming transitions. We show that for generic tilings there exists a BP transition at a nontrivial critical density, $0<\rho_c<1$. Thus, despite the presence of loops on all length scales in hyperbolic lattices, the behavior is very different from that on Euclidean lattices where the critical density is either zero or one. Furthermore, we show that the transition has a mixed character since it is discontinuous but characterized by a diverging correlation length, similarly to what happens on Bethe lattices and random graphs of constant connectivity.