Abstract
AbstractThe Eden cell growth model is a simple discrete stochastic process which produces a “blob” (aggregation of cells) in $$\mathbb {R}^d$$
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 at time t grow at a rate between $$t^{(d-1)/d}$$
t
(
d
-
1
)
/
d
and $$P_d(t)$$
P
d
(
t
)
, where $$P_d(t)$$
P
d
(
t
)
is the size of the site perimeter. Assuming a widely believed conjecture, this establishes the rate of growth of the Betti numbers in every dimension. 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.
Funder
Division of Mathematical Sciences
H2020 Marie Sklodowska-Curie Actions
Publisher
Springer Science and Business Media LLC
Subject
Computational Theory and Mathematics,Discrete Mathematics and Combinatorics,Geometry and Topology,Theoretical Computer Science
Cited by
2 articles.
订阅此论文施引文献
订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献