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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.

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    $\begingroup$ @jc: I would suggest you make your comment an answer, since that paper is the state of the art in the field. $\endgroup$
    – Igor Rivin
    Jul 8, 2012 at 4:42

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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.

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    $\begingroup$ If I recall Babson's results, depending on your biases in generation of 2-complexes you get a landscape of results of the form: generically the fundamental group tends to be either trivial, hyperbolic or free. Other types of groups tend to be rare. $\endgroup$
    – Ryan Budney
    Jul 23, 2012 at 2:00

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