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.
- Publication:
-
arXiv e-prints
- Pub Date:
- October 2019
- DOI:
- 10.48550/arXiv.1910.12803
- arXiv:
- arXiv:1910.12803
- Bibcode:
- 2019arXiv191012803D
- Keywords:
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- Mathematics - Probability;
- Mathematics - Algebraic Topology;
- Mathematics - Combinatorics;
- 60K35;
- 55U10;
- 05C80
- E-Print:
- 23 pages, 5 figures, 2 tables. This version includes several minor revisions as well as an additional Section 3.3 on continuity (with respect to the coloring probability p) of each coefficient in the limiting homotopy measure. The paper will appear in the Journal of Applied and Computational Topology