论文标题
伊甸园模型在歧管的图和镶嵌上的局部行为
Local behavior of the Eden model on graphs and tessellations of manifolds
论文作者
论文摘要
$ \ mathbb {r}^n $中的伊甸园模型如下：最初是一个单位超立方体感染的单个单元，并且每个单元均为与受感染的单元相邻的HyperCube被随机选择和感染。 Manin，Roldán和Schweinhart通过考虑可能出现在边界上的可能形状，研究了$ \ Mathbb {r}^{n} $中伊甸园模型的拓扑。特别是，它们在伊甸园模型的贝蒂数字上给出了概率下限。在本文中，我们在任何无限的，顶点交易的，有限的图形上都证明了伊甸园模型的结果：随着时间的流逝，无限的可能性很高，每一个“可能的”子图（具有轻度的“可能的“可能”含义）都发生在伊甸园模型的边界上，至少与该图形的等级概述成比例的数量。使用此过程，我们可以将有关伊甸园模型拓扑的结果扩展到非欧几里得空间，例如双曲线$ n $ - 空间和某些Riemannian歧管的通用覆盖物。
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, Roldán, 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.
