A Sharp Threshold for Spanning 2-Spheres in Random 2-Complexes

TL;DR

This paper establishes a precise threshold probability at which a Hamiltonian 2-sphere almost surely appears in a random 2-dimensional simplicial complex modeled by Linial-Meshulam, extending concepts of Hamiltonicity to higher dimensions.

Contribution

It introduces the concept of Hamiltonian d-spheres in simplicial complexes and determines a sharp threshold for their emergence in the Linial-Meshulam model.

Findings

Sharp threshold at p=√(e/γn) for Hamiltonian 2-spheres

Extension of Hamiltonian cycle concept to higher-dimensional complexes

Provides probabilistic phase transition results in random topology

Abstract

A Hamiltonian cycle in a graph is a spanning subgraph that is homeomorphic to a circle. With this in mind, it is natural to define a Hamiltonian d-sphere in a d-dimensional simplicial complex as a spanning subcomplex that is homeomorphic to a d-dimensional sphere. We consider the Linial-Meshulam model for random simplicial complexes, and prove that there is a sharp threshold at $p=\sqrt{\frac{e}{\gamma n}}$ for the appearance of a Hamiltonian $2$-sphere in a random $2$-complex, where $\gamma = 4^4/3^3$.