Homotopy Types of Random Cubical Complexes

2019/10/28 by Dowling, Kenneth, Lundberg, Erik
#05C80 #55U10 #60K35 #Algebraic Topology (math.AT) #Combinatorics (math.CO) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1910.12803

Abstract

We study the topology of a random cubical complex associated to Bernoulli site percolation on a cubical grid. We begin by establishing a limit law for homotopy types. More precisely, looking within an expanding window, we define a sequence of normalized counting measures (counting connected components according to homotopy type), and we show that this sequence of random probability measures converges in probability to a deterministic probability measure. We then investigate the dependence of the limiting homotopy measure on the coloring probability p, and our results show a qualitative change in the homotopy measure as p crosses the percolation threshold p=p_c. Specializing to the case of d=2 dimensions, we also present empirical results that raise further questions on the p-dependence of the limiting homotopy measure.

Related

• Topology of random 2-dimensional cubical complexes
• Percolation on Homology Generators in Codimension One
• Limit theorems for random cubical homology
• Patterns in Random Fractals
• Percolation in the marked stationary Random Connection Model for higher-dimensional simplicial complexes