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.

Random Geometric Complexes and Graphs on Riemannian Manifolds in the Thermodynamic Limit / Lerario, A.; Mulas, R.. - In: DISCRETE & COMPUTATIONAL GEOMETRY. - ISSN 0179-5376. - 66:3(2021), pp. 1072-1104. [10.1007/s00454-020-00238-4]

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

Lerario, A.;Mulas, R.
2021-01-01

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.
66
3
1072
1104
https://doi.org/10.1007/s00454-020-00238-4
https://arxiv.org/abs/1906.07092
Lerario, A.; Mulas, R.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11767/126842
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