Random geometric complexes

TL;DR

This paper investigates the topological properties of random geometric complexes built on i.i.d. points, revealing complex threshold behaviors in higher homology and providing asymptotic formulas using discrete Morse theory.

Contribution

It introduces higher-dimensional analogues of connectivity results and identifies multiple thresholds for homology in random geometric complexes, employing discrete Morse theory.

Findings

Higher homology H_k is not monotone for k > 0.

Existence of two thresholds for each k where homology appears and disappears.

Asymptotic formulas for Betti numbers in sparse regimes.

Abstract

We study the expected topological properties of Cech and Vietoris-Rips complexes built on i.i.d. random points in R^d. We find higher dimensional analogues of known results for connectivity and component counts for random geometric graphs. However, higher homology H_k is not monotone when k > 0. In particular for every k > 0 we exhibit two thresholds, one where homology passes from vanishing to nonvanishing, and another where it passes back to vanishing. We give asymptotic formulas for the expectation of the Betti numbers in the sparser regimes, and bounds in the denser regimes. The main technical contribution of the article is in the application of discrete Morse theory in geometric probability.