Stationary Eden model on groups

TL;DR

This paper studies stationary Eden models on half-plane lattices and Cayley graphs, proving that all trees are finite almost surely under weak conditions, extending previous results on related growth models.

Contribution

It introduces a generalized approach to Eden models on various graphs and proves finiteness of trees using ergodic and mass transport principles.

Findings

All trees are finite almost surely in the considered models.

The results extend to graphs of the form G×Z_+ where G is a Cayley graph.

Generalizes known results on the two-type Richardson model.

Abstract

We consider two stationary versions of the Eden model, on the upper half planar lattice, resulting in an infinite forest covering the half plane. Under weak assumptions on the weight distribution and by relying on ergodic theorems, we prove that almost surely all trees are finite. Using the mass transport principle, we generalize the result to Eden model in graphs of the form $G\times\mathbb{Z}_+$, where $G$ is a Cayley graph. This generalizes certain known results on the two-type Richardson model, in particular of Deijfen and H\"aggstr\"om in 2007.