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

TL;DR

This paper identifies the precise probability threshold at which the top homology group of random d-dimensional complexes almost surely vanishes as the number of vertices grows, extending understanding of topological phase transitions.

Contribution

It establishes the exact threshold function for the vanishing of the top homology group in random d-complexes with full (d-1)-skeletons, for all dimensions d.

Findings

Determines the threshold function for homology vanishing.

Provides a comprehensive analysis for all d ≥ 1.

Enhances understanding of topological phase transitions in random complexes.

Abstract

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 $d\geq 1$.