# Random geometric complexes and graphs on Riemannian manifolds in the thermodynamic limit

## TL;DR

This paper studies the topological and spectral properties of random geometric complexes and graphs on Riemannian manifolds, showing convergence of measures in the thermodynamic limit.

## Contribution

It proves convergence results for the distribution of component types and spectral measures of random geometric complexes and graphs on Riemannian manifolds.

## Key findings

- Normalized counting measure of components converges to a deterministic measure
- Spectral measure of the Laplace operator converges in the case of random geometric graphs
- Results hold in the thermodynamic limit for large complexes and graphs

## Abstract

We investigate some topological properties of random geometric complexes and random geometric graphs on Riemannian manifolds in the thermodynamic limit. In particular, for random geometric complexes we prove that the normalized counting measure of connected components, counted according to isotopy type, converges in probability to a deterministic measure. More generally, we also prove similar convergence results for the counting measure of types of components of each $k$-skeleton of a random geometric complex. As a consequence, in the case of the $1$-skeleton (i.e. for random geometric graphs) we show that the empirical spectral measure associated to the normalized Laplace operator converges to a deterministic measure.