Homotopy Types of Random Cubical Complexes

TL;DR

This paper analyzes the topological properties of random cubical complexes generated by percolation on grids, establishing a limit law for homotopy types and exploring how these types change with percolation probability.

Contribution

It introduces a limit law for the distribution of homotopy types in random cubical complexes and examines how this distribution varies with percolation probability, especially around the critical threshold.

Findings

Homotopy type distribution converges to a deterministic measure as the window expands.

The limiting homotopy measure qualitatively changes at the percolation threshold p_c.

Empirical results in 2D suggest further p-dependence questions for the homotopy measure.

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.