{"paper":{"title":"Thresholds for vanishing of `Isolated' faces in random \\v{C}ech and Vietoris-Rips complexes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.PR","authors_text":"D. Yogeshwaran, Srikanth K. Iyer","submitted_at":"2018-02-22T18:23:52Z","abstract_excerpt":"We study combinatorial connectivity for two models of random geometric complexes. These two models - \\v{C}ech and Vietoris-Rips complexes - are built on a homogeneous Poisson point process of intensity $n$ on a $d$-dimensional torus using balls of radius $r_n$. In the former, the $k$-simplices/faces are formed by subsets of $(k+1)$ Poisson points such that the balls of radius $r_n$ centred at these points have a mutual interesection and in the latter, we require only a pairwise intersection of the balls. Given a (simplicial) complex (i.e., a collection of $k$-simplices for all $k \\geq 1$), we "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1802.08224","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"}