论文标题
伊甸园模型的拓扑和局部几何形状
Topology and local geometry of the Eden model
论文作者
论文摘要
伊甸园细胞生长模型是一个简单的离散随机过程，在$ \ mathbb {r}^d $中产生“斑点”，从常规网格中的一个立方体开始，并在每个时间步骤随机均匀地添加一个相邻的立方体。该过程已用作聚集，肿瘤和细菌菌落生长以及伤口愈合等自然过程的模型。在这里，我们研究了所得结构的拓扑结构和局部几何形状，为贝蒂数字建立了渐近界。我们的主要结果是，贝蒂的数量以猜想的地点周长增长率与现场周边的实际生长速率之间的速度增长。我们还介绍了计算实验的结果，内容涉及几何和拓扑的更细微方面，例如持续的同源性和孔形状的分布。
The Eden cell growth model is a simple discrete stochastic process which produces a "blob" in $\mathbb{R}^d$: start with one cube in the regular grid, and at each time step add a neighboring cube uniformly at random. This process has been used as a model for the growth of aggregations, tumors, and bacterial colonies and the healing of wounds, among other natural processes. Here, we study the topology and local geometry of the resulting structure, establishing asymptotic bounds for Betti numbers. Our main result is that the Betti numbers grow at a rate between the conjectured rate of growth of the site perimeter and the actual rate of growth of the site perimeter. We also present the results of computational experiments on finer aspects of the geometry and topology, such as persistent homology and the distribution of shapes of holes.
