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Law of large numbers for Betti numbers of homogeneous and spatially independent random simplicial complexes
Author(s) -
Kanazawa Shu
Publication year - 2022
Publication title -
random structures and algorithms
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.314
H-Index - 69
eISSN - 1098-2418
pISSN - 1042-9832
DOI - 10.1002/rsa.21015
Subject(s) - betti number , mathematics , simplicial complex , random graph , central limit theorem , simplicial homology , abstract simplicial complex , simplicial approximation theorem , combinatorics , homogeneous , limit (mathematics) , scaling limit , discrete mathematics , graph , scaling , pure mathematics , mathematical analysis , simplicial set , statistics , homotopy , homotopy category , geometry
Abstract The Linial–Meshulam complex model is a natural higher dimensional analog of the Erdős–Rényi graph model. In recent years, Linial and Peled established a limit theorem for Betti numbers of Linial–Meshulam complexes with an appropriate scaling of the underlying parameter. The present article aims to extend that result to more general random simplicial complex models. We introduce a class of homogeneous and spatially independent random simplicial complexes, including the Linial–Meshulam complex model and the random clique complex model as special cases, and we study the asymptotic behavior of their Betti numbers. Moreover, we obtain the convergence of the empirical spectral distributions of their Laplacians. A key element in the argument is the local weak convergence of simplicial complexes. Inspired by the work of Linial and Peled, we establish the local weak limit theorem for homogeneous and spatially independent random simplicial complexes.