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

A. Lerario; R. Mulas

doi:10.1007/s00454-020-00238-4

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

A. Lerario
, R. Mulas

Research output: Contribution to Journal › Article › Academic › peer-review

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.

Original language	English
Pages (from-to)	1072-1104
Number of pages	33
Journal	Discrete and Computational Geometry
Volume	66
Issue number	3
Early online date	31 Aug 2020
DOIs	https://doi.org/10.1007/s00454-020-00238-4
Publication status	Published - Oct 2021
Externally published	Yes

Funding

Open access funding provided by Scuola Internazionale Superiore di Studi Avanzati - SISSA within the CRUI-CARE Agreement. We are grateful to Bernd Sturmfels, because without him this paper would not exist. We are grateful to Fabio Cavalletti, Jürgen Jost, Matthew Kahle, Erik Lundberg, Leo Mathis, and Michele Stecconi for helpful comments, discussions, and forbidden graphs. We are grateful to the anonymous referees for the constructive comments.

Funders
Scuola Internazionale Superiore di Studi Avanzati

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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.",

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AB - 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.

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

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