Interacting growth processes and invariant percolation

TL;DR

This paper explores the connection between reversible growth processes and invariant percolation, introducing models of interacting branching random walks that relate survival to infinite clusters in invariant percolation on trees.

Contribution

It presents a conceptual framework linking growth processes with invariant percolation and introduces new models of interacting branching random walks with potential for broader generalizations.

Findings

Survival of growth processes corresponds to infinite clusters in invariant percolation.

Models include truncated and competing branching random walks.

Framework allows for generalizations to other reversible growth processes.

Abstract

The aim of this paper is to underline the relation between reversible growth processes and invariant percolation. We present two models of interacting branching random walks (BRWs), truncated BRWs and competing BRWs, where survival of the growth process can be formulated as the existence of an infinite cluster in an invariant percolation on a tree. Our approach is fairly conceptual and allows generalizations to a wider set of "reversible" growth processes.