On simple connectivity of random 2-complexes

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 $p\approx n^{-1\mathrm{/}2}$. In this paper, we show that this threshold probability is at most $p\le (\gamma n{)}^{-1\mathrm{/}2}$, where $\gamma = 4^4/3^3$, and conjecture that this threshold is sharp. In fact, we show that $p=(\gamma n{)}^{-1\mathrm{/}2}$ 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.