{"paper":{"title":"Limit theorems for process-level Betti numbers for sparse, critical, and Poisson regimes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Andrew Thomas, Takashi Owada","submitted_at":"2018-09-15T19:16:57Z","abstract_excerpt":"The objective of this study is to examine the asymptotic behavior of Betti numbers of \\v{C}ech complexes treated as stochastic processes and formed from random points in the $d$-dimensional Euclidean space $\\mathbb{R}^d$. We consider the case where the points of the \\v{C}ech complex are generated by a Poisson process with intensity $nf$ for a probability density $f$. We look at the cases where the behavior of the connectivity radius of \\v{C}ech complex causes simplices of dimension greater than $k+1$ to vanish in probability, the so-called sparse and Poisson regimes, as well when the connectiv"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.05758","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}