# Topologies of random geometric complexes on Riemannian manifolds in the thermodynamic limit

## TL;DR

This paper studies the topological properties of random geometric complexes on Riemannian manifolds in the thermodynamic limit, establishing universal laws and the convergence of homotopy type distributions.

## Contribution

It proves the existence of universal limit laws for the topology of complexes and characterizes the support of the limiting measure in the thermodynamic regime.

## Key findings

- Convergence of normalized homotopy type counts to a deterministic measure
- Support of the limiting measure includes all Euclidean homotopy types
- Universal laws apply to complexes on Riemannian manifolds

## 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.