Homotopy of random simplicial complexes - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T06:53:55Z http://mathoverflow.net/feeds/question/101598 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/101598/homotopy-of-random-simplicial-complexes Homotopy of random simplicial complexes Pinying 2012-07-07T19:13:02Z 2012-07-23T01:42:37Z <p>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 <a href="http://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93R%C3%A9nyi_model" rel="nofollow">Erdős–Rényi</a> model $G(n,p)$ for generating random graphs. </p> <p>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 <a href="http://www.math.ias.edu/~mkahle/Betti.pdf" rel="nofollow">this</a> paper.</p> <p>I want to ask</p> <blockquote> <p>What is known about the properties of the fundamental group (or higher homotopy groups) of random simplicial complexes?</p> </blockquote> <p>If there is a good reference, that would be enough. I can not find one on google. Thank you for your time.</p> http://mathoverflow.net/questions/101598/homotopy-of-random-simplicial-complexes/101626#101626 Answer by jc for Homotopy of random simplicial complexes jc 2012-07-08T07:14:15Z 2012-07-23T01:42:37Z <p>Babson, Hoffman, and Kahle have written <a href="http://arxiv.org/abs/0711.2704" rel="nofollow">a paper</a> 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.</p> <p>Babson has just written <a href="http://arxiv.org/abs/1207.5028" rel="nofollow">a paper</a> on the fundamental groups of clique complexes of Erdős–Rényi random graphs using similar techniques. </p>