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.
"Interacting growth processes and invariant percolation." Ann. Appl. Probab. 25 (1) 268 - 286, February 2015. https://doi.org/10.1214/13-AAP995