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](http://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93R%C3%A9nyi_model) 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](http://www.math.ias.edu/~mkahle/Betti.pdf) paper.


I want to ask



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> What is known about the properties of the fundamental group (or higher homotopy groups) of random simplicial complexes?
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If there is a good reference, that would be enough. I can not find one on google. Thank you for your time.



Homotopy of random simplicial complexes
Babson, Hoffman, and Kahle have written [a paper](http://arxiv.org/abs/0711.2704) 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](http://arxiv.org/abs/1207.5028) on the fundamental groups of clique complexes of Erdős–Rényi random graphs using similar techniques. 


