On simple connectivity of random 2-complexes

TL;DR

This paper refines the threshold probability for simple connectivity in random 2-complexes, showing it is at most proportional to n^{-1/2} and providing evidence it is a sharp threshold, using combinatorial enumeration techniques.

Contribution

It establishes an improved upper bound for the connectivity threshold and conjectures its sharpness, introducing a new perspective on the topology of random complexes.

Findings

Threshold probability for simple connectivity is at most (rac{4^4}{3^3} n)^{-1/2}

Sharp threshold identified for cycles of length 3 bounding disk-like subcomplexes

Uses Poisson paradigm and Tutte's enumeration of planar triangulations

Abstract

The fundamental group of the $2$-dimensional Linial-Meshulam random simplicial complex $Y_2(n,p)$ was first studied by Babson, Hoffman and Kahle. They proved that the threshold probability for simple connectivity of $Y_2(n,p)$ is about $p\approx n^{-1/2}$. In this paper, we show that this threshold probability is at most $p\le (\gamma n)^{-1/2}$, where $\gamma = 4^4/3^3$, and conjecture that this threshold is sharp. In fact, we show that $p=(\gamma n)^{-1/2}$ is a sharp threshold probability for the stronger property that every cycle of length $3$ is the boundary of a subcomplex of $Y_2(n,p)$ that is homeomorphic to a disk. Our proof uses the Poisson paradigm, and relies on a classical result of Tutte on the enumeration of planar triangulations.