Random Simplicial Complexes: Around the Phase Transition

Abstract

This article surveys some of the work done in recent years on random simplicial complexes. We mostly consider higher-dimensional analogs of the well known phase transition in G(n, p) theory that occurs at p = 1 n . Our main objective is to provide a more streamlined and unified perspective of some of the papers in this area.

Law of large numbers for Betti numbers of homogeneous and spatially independent random simplicial complexes

“…In this section, we consider homogeneous and spatially independent random subcomplexes of △ n , and we study their local weak convergence as n tends to infinity. To state the theorem, we describe a higher dimensional generalization of the Galton-Watson tree with Poisson offspring distribution (see also [3], section 3; [22], section 3).…”

“…The local weak convergence is critical for estimating the asymptotic behavior of the Betti numbers. This type of approach has been studied in various contexts [2,3,10,21,22,25]. Inspired by those studies, especially the formulation in [22], we establish the local weak limit theorem for homogeneous and spatially independent random simplicial complexes (Theorem 13), where a higher dimensional generalization of the Poisson branching process arises as a universal limiting object.…”

“…This type of approach has been studied in various contexts [2,3,10,21,22,25]. Inspired by those studies, especially the formulation in [22], we establish the local weak limit theorem for homogeneous and spatially independent random simplicial complexes (Theorem 13), where a higher dimensional generalization of the Poisson branching process arises as a universal limiting object. Consequently, we also obtain the convergence of the empirical spectral distributions of their Laplacians (Theorem 24 (2)).…”

Freeness of the random fundamental group

“…It is proved in [2] that such a series of elementary collapses is not possible for c > γ 2 . Furthermore, [11] points out that in the regime γ 2 < c < c 2 , Y (n, c/n) is far from being 2-collapsible in the sense that a constant fraction of the faces must be deleted to arrive at a complex which is 2-collapsible. Thus Y (n, c/n) in the regime γ 2 < c < c 2 if (2) holds would be homotopy equivalent to a wedge of circles, but not via the same type of homotopy equivalence which exists for smaller values of c.…”

Tri-partitions and Bases of an Ordered Complex

“…For example, does the tri-partition lead to a fast algorithm for constructing harmonic cycles, that is, whose Laplacian is zero? • Can the tri-partitions be used to shed light on the stochastic properties of simplicial complexes as studied in [15]?…”