Topology and local geometry of the Eden model
Abstract
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.
 Publication:

arXiv eprints
 Pub Date:
 May 2020
 arXiv:
 arXiv:2005.12349
 Bibcode:
 2020arXiv200512349M
 Keywords:

 Mathematics  Probability;
 Mathematical Physics;
 Mathematics  Algebraic Topology;
 Mathematics  Combinatorics;
 82B43;
 62R40;
 60K35;
 55N31;
 05B50
 EPrint:
 31 pages, 10 figures, 5 tables