On simple connectivity of random 2-complexes

2018/06/08 by Luria, Zur, Peled, Yuval · 1 citation
#Combinatorics (math.CO) #FOS: Mathematics #Geometric Topology (math.GT) #Probability (math.PR)

paper · doi:10.48550/arxiv.1806.03351

Abstract

The fundamental group of the 2-dimensional Linial-Meshulam random simplicial complex Y₂(n,p) was first studied by Babson, Hoffman and Kahle. They proved that the threshold probability for simple connectivity of Y₂(n,p) is about p≈ n^-1/2. In this paper, we show that this threshold probability is at most p≤ (γn)^-1/2, where γ= 4⁴/3³, and conjecture that this threshold is sharp. In fact, we show that p=(γn)^-1/2 is a sharp threshold probability for the stronger property that every cycle of length 3 is the boundary of a subcomplex of Y₂(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.

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