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Homotopy types of random cubical complexes. (English) Zbl 1490.60267

Summary: 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.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
55U10 Simplicial sets and complexes in algebraic topology
05C80 Random graphs (graph-theoretic aspects)

Software:

Perseus
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Full Text: DOI arXiv

References:

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