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 connected geometric complexes of the same dimension as the manifold.
Topologies of Random Geometric Complexes on Riemannian Manifolds in the Thermodynamic Limit / Auffinger, Antonio; Lerario, Antonio; Lundberg, Erik. - In: INTERNATIONAL MATHEMATICS RESEARCH NOTICES. - ISSN 1073-7928. - 2021:20(2021), pp. 1-36. [10.1093/imrn/rnaa050]
Topologies of Random Geometric Complexes on Riemannian Manifolds in the Thermodynamic Limit
Lerario, Antonio;
2021-01-01
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 connected geometric complexes of the same dimension as the manifold.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.