The threshold function for vanishing of the top homology group of random $d$-complexes

By Dmitry N. Kozlov. 2009

50 citations recorded by the source corpus.

For positive integers $n$ and $d$, and the probability function $0\leq p(n)\leq 1$, we let $Y_{n,p,d}$ denote the probability space of all at most $d$-dimensional simplicial complexes on $n$ vertices, which contain the full $(d-1)$-dimensional skeleton, and whose $d$-simplices appear with probability $p(n)$. In this paper we determine the threshold function for vanishing of the top homology group in $Y_{n,p,d}$, for all $d\geq 1$.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/cc74b9ed6e28e6f77de1162f37b18daa1cf592cc

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In Computational Theory and Mathematics

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Citations in the corpus

Spectral Gaps of Random Graphs and Applications (2012, background): “al analogue of the Erdo˝s-R´enyi threshold for connectivity of the random graph G(n,p) [12]. This paper inspired several other articles which have explored the topology of random simplicial complexes [2, 23, 27, 24, 1]. Random simplicial complexes have also been studied for their expansion propertie”
Topology of random simplicial complexes: a survey (2013, background): “Since the influential paper [41], random complexes and their topological properties have continued to be explored by several teams of researchers [43, 7, 39, 4, 5, 35].”
The existence of designs (2014, result): “ operators in (pseudo)random simplicial complexes may merit further study from a topological point of view; it is reminiscent of recent results on the topology of random simplicial complexes (see e.g [2, 26]). There are many potential directions of generalisation suggested by known results on graph ”
Sharp vanishing thresholds for cohomology of random flag complexes (2012, background): “ribed in the introduction – or you might say that it is sharp on one side and not on the other. We would expect that the same thing is true for appearance of nontrivial k-cycles in X(n,p). See Kozlov [22], Cohen et. al. [10], and Aronshtam–Linial [3] for closely related work on the threshold for the”
Coboundary expanders (2010, background): “These complexes have well-studied topological and geometric behavior ([4], [15], [3]), and they provide a template for studying a broader family of random complexes.”