Random Geometric Complexes and Graphs on Riemannian Manifolds in the Thermodynamic Limit
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Random Geometric Complexes and Graphs on Riemannian Manifolds in the Thermodynamic Limit Antonio Lerario1
· Raffaella Mulas2
Received: 28 September 2019 / Revised: 23 April 2020 / Accepted: 21 July 2020 © The Author(s) 2020
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. Keywords Random graphs · Graph Laplacian · Random geometric complexes Mathematics Subject Classification 05C62 · 05C80 · 15B52
1 Introduction 1.1 Random Geometric Complexes The subject of random geometric complexes has recently attracted a lot of attention, with a special focus on the study of expectation of topological properties of these com-
Editor in Charge: János Pach Antonio Lerario [email protected] Raffaella Mulas [email protected] 1
SISSA, Trieste, Italy
2
Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany
123
Discrete & Computational Geometry
plexes [5,6,31,39,43,53]1 (e.g. number of connected components, or, more generally, Betti numbers). In a recent paper [1], Auffinger, Lerario, and Lundberg have imported methods from [38,48] for the study of finer properties of these random complexes, namely the distribution of the homotopy types of the connected components of the complex. Before moving to the content of the current paper, we discuss the main ideas from [1] and introduce some terminology. Let (M, g) be a compact, Riemannian manifold of dimension m. We normalize the metric g in such a way that vol M = 1. ˆ We denote by B(x, r ) ⊂ M the Riemannian ball centered at x of radius r > 0 and we construct a random M-geometric complex in the thermodynamic regime as follows. We let { p1 , . . . , pn } be a set of points independently sampled from the uniform distribution on M, we fix a positive number α > 0, and we consider Un :=
n
ˆ pk , r ) where r := αn −1/m . B(
(1.1)
k=1
The choice of such r is what defines the so-called critical or thermodynamic regime2 and it is the regime where topology is the richest [1,31]. We say that Un is a random M-geometric complex; the name is motivated by the fact that, for n large enough, Un ˇ is homotopy equivalent to its Cech complex, as we shall see in Lemma 2.4 below. Auffinger, Lerario, and Lundberg [1] proved that, in the case when vol M = 1, the normalized counting measure of connected components of such complexes, counted according to homotopy type, converges in probability to a dete
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