Local behavior of the Eden model on graphs and tessellations of manifolds

TL;DR

This paper extends the understanding of the Eden model's boundary topology from Euclidean spaces to general graphs and non-Euclidean spaces, demonstrating that all feasible boundary shapes occur frequently over time.

Contribution

It proves that on infinite, vertex-transitive graphs, the Eden model's boundary contains all possible subgraphs with high probability, generalizing previous Euclidean results to broader spaces.

Findings

All feasible boundary subgraphs occur frequently over time.

Results extend to hyperbolic spaces and Riemannian manifold covers.

Provides probabilistic lower bounds on boundary topology.

Abstract

The Eden Model in $\mathbb{R}^n$ constructs a blob as follows: initially a single unit hypercube is infected, and each second a hypercube adjacent to the infected ones is selected randomly and infected. Manin, Rold\'{a}n, and Schweinhart investigated the topology of the Eden model in $\mathbb{R}^{n}$ by considering the possible shapes which can appear on the boundary. In particular, they give probabilistic lower bounds on the Betti numbers of the Eden model. In this paper, we prove analogous results for the Eden model on any infinite, vertex-transitive, locally finite graph: with high probability as time goes to infinity, every "possible" subgraph (with mild conditions on what "possible" means) occurs on the boundary of the Eden model at least a number of times proportional to an isoperimetric profile of the graph. Using this, we can extend the results about the topology of the Eden model to non-Euclidean spaces, such as hyperbolic $n$-space and universal covers of certain Riemannian manifolds.