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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<jats:italic>k<\/jats:italic>-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.<\/jats:p>","DOI":"10.1007\/s00454-020-00238-4","type":"journal-article","created":{"date-parts":[[2020,8,31]],"date-time":"2020-08-31T20:03:28Z","timestamp":1598904208000},"page":"1072-1104","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Random Geometric Complexes and Graphs on Riemannian Manifolds in the Thermodynamic Limit"],"prefix":"10.1007","volume":"66","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5986-1353","authenticated-orcid":false,"given":"Antonio","family":"Lerario","sequence":"first","affiliation":[]},{"given":"Raffaella","family":"Mulas","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2020,8,31]]},"reference":[{"key":"238_CR1","unstructured":"Auffinger, A., Lerario, A., Lundberg, E.: Topologies of random geometric complexes on Riemannian manifolds in the thermodynamic limit (2018). arXiv:1812.09224 (to appear in IMRN)"},{"issue":"5439","key":"238_CR2","doi-asserted-by":"publisher","first-page":"509","DOI":"10.1126\/science.286.5439.509","volume":"286","author":"A-L Barab\u00e1si","year":"1999","unstructured":"Barab\u00e1si, A.-L., Albert, R.: Emergence of scaling in random networks. 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