Random geometric complexes in the thermodynamic regime

Abstract

We consider the topology of simplicial complexes with vertices the points of a random point process and faces determined by distance relationships between the vertices. In particular, we study the Betti numbers of these complexes as the number of vertices becomes large, obtaining limit theorems for means, strong laws, concentration inequalities and central limit theorems. As opposed to most prior papers treating random complexes, the limit with which we work is in the so-called 'thermodynamic' regime (which includes the percolation threshold) in which the complexes become very large and complicated, with complex homology characterised by diverging Betti numbers. The proofs combine probabilistic arguments from the theory of stabilizing functionals of point processes and topological arguments exploiting the properties of Mayer-Vietoris exact sequences. The Mayer-Vietoris arguments are crucial, since homology in general, and Betti numbers in particular, are global rather than local phenomena, and most standard probabilistic arguments are based on the additivity of functionals arising as a consequence of locality.

Random Čech complexes on Riemannian manifolds

“…We will refer to these generalizations as random combinatorial complexes (see [26] for a survey). It turns out that the Erdős -Rényi threshold for connectivity can be generalized to that of "homological connectivity," where the higher homology groups H k become trivial.In parallel to the study of combinatorial complexes, a line of research was established for random geometric complexes [3,5,24,42,43]. This type of complexes generalizes the model of the random geometric graph G(n, r) (introduced by Gilbert [18]), where vertices are placed at random in a metric-measure space, and edges are included based on proximity [38].…”

A remark on the convergence of Betti numbers in the thermodynamic regime

“…For homogeneous Poisson point processes, the following law of large numbers for Betti numbers was established in [9]. Let P(λ) be a homogeneous Poisson point process on R d with density λ > 0.…”

“…provided that the density function f has compact, convex support and that on the support of f, it is bounded both below and above [9,Theorem 4.6]. A remaining problem is to describe the exact limiting behaviour of the expected values of the Betti numbers.…”

“…The first one is the nearly additive property of Betti numbers that was used in [9] to study Betti numbers ofČech complexes built on stationary point processes. The second one is the property that binomial point processes behave locally like a homogeneous Poisson point process.…”

Topologies of Random Geometric Complexes on Riemannian Manifolds in the Thermodynamic Limit

“…Yogeshwaran, Subag, and Adler [YSA17] have established limit laws in the thermodynamic regime for Betti numbers of random geometric complexes built over Poisson point processes in Euclidean space. Their results include limit theorems for expectations as well as concentration inequalities and central limit theorems.…”