Homotopy of random simplicial complexes

Question

A random graph on $n$ vertices is defined by selectiung the edges according to some probability distribution, the simplest case being the one where the edge between any two vertices exists with probability $p = \frac{1}{2}$. I believe this is the Erdős–Rényi model $G(n,p)$ for generating random graphs.

Similarly, in higher dimensions we can construct random simplicial complexes on $n$ vertices in many ways. One such method is as follows: fix a top dimension $d$, and now define the random simplicial model $S_d(n,p)$ where each $d$ simplex spanning any $d+1$ vertices exists with probability $p$. Some work has been done investigating the homology of such complexes in limiting cases, see for example this paper.

I want to ask

What is known about the properties of the fundamental group (or higher homotopy groups) of random simplicial complexes?

If there is a good reference, that would be enough. I can not find one on google. Thank you for your time.

Answer 1

Babson, Hoffman, and Kahle have written a paper on fundamental groups of random 2-complexes. They worked with the Linial-Meshulam model whereby you begin with a complete graph on $n$ vertices and then add independently uniformly random 2-simplices.

Babson has just written a paper on the fundamental groups of clique complexes of Erdős–Rényi random graphs using similar techniques.