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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.
There have been several recent articles studying homology of various types of random simplicial complexes. Several theorems have concerned thresholds for vanishing of homology, and in some cases expectations of the Betti numbers. However little seems
We correct the proofs of the main theorems in our paper Limit theorems for Betti numbers of random simplicial complexes.
We implement an algorithm RSHT (Random Simple-Homotopy) to study the simple-homotopy types of simplicial complexes, with a particular focus on contractible spaces and on finding substructures in higher-dimensional complexes. The algorithm combines el
Random abstract simplicial complex representation provides a mathematical description of wireless networks and their topology. In order to reduce the energy consumption in this type of network, we intend to reduce the number of network nodes without
We investigate the topologies of random geometric complexes built over random points sampled on Riemannian manifolds in the so-called thermodynamic regime. We prove the existence of universal limit laws for the topologies; namely, the random normaliz