Percolation on an infinitely generated group

TL;DR

This paper presents a long-range Bernoulli percolation model on a complex group where all clusters are finite, exploring its properties, definitions, and phase transitions, expanding understanding of percolation on non-quasi-isometric groups.

Contribution

It introduces a novel percolation process on a non-quasi-isometric group with finite clusters, analyzing its properties and phase transitions.

Findings

Clusters are almost surely finite for all parameters

The model admits multiple equivalent definitions

Identification of certain phase transitions

Abstract

We give an example of a long range Bernoulli percolation process on a group non-quasi-isometric with $\mathbb{Z}$, in which clusters are almost surely finite for all values of the parameter. This random graph admits diverse equivalent definitions, and we study their ramifications. We also study its expected size and point out certain phase transitions.