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.

中文翻译：

---

Eden 模型在图和流形镶嵌上的局部行为

$\mathbb{R}^n$ 中的伊甸园模型构造一个 blob 如下：最初一个单位超立方体被感染，每秒随机选择一个与被感染超立方体相邻的超立方体并被感染。Manin、Rold\'{a}n 和 Schweinhart 通过考虑可能出现在边界上的形状，研究了 $\mathbb{R}^{n}$ 中 Eden 模型的拓扑结构。特别是，它们给出了 Eden 模型的 Betti 数的概率下限。在本文中，我们证明了 Eden 模型在任何无限的、顶点传递的、局部有限的图上的类似结果：随着时间趋于无穷大，每个“可能的”子图（对“可能”的含义有温和的条件）发生的概率很高在 Eden 模型的边界上，至少与图形的等周轮廓成比例的次数。