The threshold for integer homology in random d-complexes

TL;DR

This paper determines the precise threshold for the vanishing of the (d-1)-dimensional homology in random d-complexes, showing it occurs around 80d log n / n, confirming a longstanding conjecture.

Contribution

It establishes the exact asymptotic threshold for homology vanishing in random d-complexes, resolving a question posed by Linial and Meshulam in 2003.

Findings

Threshold for homology vanishing is less than 80d log n / n

Bound is tight up to a constant factor

Provides a precise phase transition point for homology in random complexes

Abstract

Let Y ~ Y_d(n,p) denote the Bernoulli random d-dimensional simplicial complex. We answer a question of Linial and Meshulam from 2003, showing that the threshold for vanishing of homology H_{d-1}(Y; Z) is less than 80d log n / n. This bound is tight, up to a constant factor.