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

Abstract

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 normalized counting measure of connected components (counted according to homotopy type) is shown to converge in probability to a deterministic probability measure. Moreover, we show that the support of the deterministic limiting measure equals the set of all homotopy types for Euclidean geometric complexes of the same dimension as the manifold. arXiv:1812.09224v1 [math.PR]

Direction distribution for nodal components of random band-limited functions on surfaces

“…provided we have the restriction (4.4) and a < 1, where we have used some well-known formulas for inverses of small perturbations of the identity matrix. This in turn implies that the inverse function G −1 ψ exists on B(u 0 , b), therefore establishing claim (2). Hence any zero, if it exists, lying inside of B(u 0 , b) must be unique and isolated.…”

Random Geometric Complexes and Graphs on Riemannian Manifolds in the Thermodynamic Limit

“…number of connected components, or, more generally, Betti numbers). In a recent paper [1], Auffinger, Lerario, and Lundberg have imported methods from [38,48] for the study of finer properties of these random complexes, namely the distribution of the homotopy types of the connected components of the complex. Before moving to the content of the current paper, we discuss the main ideas from [1] and introduce some terminology.…”

“…. , p n } be a set of points independently sampled from the uniform distribution on M, we fix a positive number α > 0, and we consider The choice of such r is what defines the so-called critical or thermodynamic regime 2 and it is the regime where topology is the richest [1,31]. We say that U n is a random M-geometric complex; the name is motivated by the fact that, for n large enough, U n is homotopy equivalent to itsČech complex, as we shall see in Lemma 2.4 below.…”

“…We say that U n is a random M-geometric complex; the name is motivated by the fact that, for n large enough, U n is homotopy equivalent to itsČech complex, as we shall see in Lemma 2.4 below. Auffinger, Lerario, and Lundberg [1] proved that, in the case when vol M = 1, the normalized counting measure of connected components of such complexes, counted according to homotopy type, converges in probability to a deterministic measure. That is,˜…”

“…We now move to the content of the current paper. Our first goal is to include the results of [1] into a more general framework which allows to make even more refined counts (e.g. according to the type of the embedding of the components, or on the structure of their skeleta, or on the property of containing a given motif 3 ).…”