Homotopy types of random cubical complexes

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.