Threshold conjecture for simple connectivity in random 2-complexes

Let $Y_2(n,p)$ be the Linial–Meshulam random $2$-complex on $n$ vertices, where each possible $2$-simplex is included independently with probability $p$, and let $\gamma:=4^4/3^3$. A cycle is triangulated in a simplicial complex if it is the boundary of a triangulation of the $2$-dimensional disk that is a subcomplex. Simple-connectivity threshold conjecture. The sharp threshold probability for simple connectivity of $Y_2(n,p)$ is

$$p=(\gamma n)^{-1/2}.$$

The preceding theorem establishes that, for $p=c/\sqrt n$ with $c>1/\sqrt\gamma$, the random complex is asymptotically almost surely simply connected, while the conjecture identifies the corresponding sharp threshold.