Limit theory of sparse random geometric graphs in high dimensions

TL;DR

This paper investigates the topological and geometric properties of high-dimensional sparse random geometric graphs, providing asymptotic results and limit theorems for various functionals including Betti numbers and subgraph counts.

Contribution

It introduces new asymptotic and limit theorems for topological and geometric functionals of sparse high-dimensional random geometric graphs, extending to persistent Betti numbers.

Findings

Moment asymptotics established

Functional central limit theorems proven

Poisson approximation results obtained

Abstract

We study topological and geometric functionals of $l_\infty$-random geometric graphs on the high-dimensional torus in a sparse regime, where the expected number of neighbors decays exponentially in the dimension. More precisely, we establish moment asymptotics, functional central limit theorems and Poisson approximation theorems for certain functionals that are additive under disjoint unions of graphs. For instance, this includes simplex counts and Betti numbers of the Rips complex, as well as general subgraph counts of the random geometric graph. We also present multi-additive extensions that cover the case of persistent Betti numbers of the Rips complex.