Random simplicial complexes

This is a post about Matt Kahle’s cool paper “Sharp vanishing thresholds for cohomology of random flag complexes,” which has just been accepted in the Annals.

The simplest way to make a random graph is to start with n vertices and then, for each pair (i,j) independently, put an edge between vertices i and j with probability p.  That’s called the Erdös-Rényi graph G(n,p), after the two people who first really dug into its properties.  What’s famously true about Erdös-Rényi graphs is that there’s a sharp threshold for connectness.  Imagine n being some fixed large number and p varying from 0 to 1 along a slider.  When p is very small relative to n, G(n,p) is very likely to be disconnected; in fact, if

there is very likely to be an isolated vertex, which makes G(n,p) disconnected all by itself.

On the other hand, if

then G(n,p) is almost surely connected!  In other words, the probability of connectedness “snaps” from 0 to 1 as you cross the barrier p = (log n)/n.  Of course, there are lots of other interesting questions you can ask — what exactly happens very near the “phase transition”?  For p < (log n)/n, what do the components look like?  (Answer:  for some range of p there is, with probability 1, a single “giant component” much larger than all others.  For instance, when p = 1/n the giant component has size around n^{2/3}.)

I think it’s safe to say that the Erdös-Rényi graph is the single most-studied object in probabilistic combinatorics.

But Kahle asked a very interesting question about it that was completely new to me.  Namely:  what if you consider the flag complex X(n,p), a simplicial complex whose k-simplices are precisely the k-cliques in G(n,p)?  X(n,p) is connected precisely when G(n,p) is, so there’s nothing new to say from that point of view.  But, unlike the graph, the complex has lots of interesting higher homology groups!  The connectedness threshold says that dim H_0(X(n,p)) is 1 above some sharp threshold and larger below it.  What Kahle proves is that a similar threshold exists for all the homology.  Namely, for each k there’s a range (bounded approximately by and $(log n / n)^{1/(k+1)}$) such that H_k(X(n,p)) vanishes when p is outside the range, but not when p is inside the range!  So there are two phase transitions; first, H^k appears, then it disappears.  (If I understand correctly, there’s a narrow window where two consecutive Betti numbers are nonzero, but most of the time there’s only one nonzero Betti number.)  Here’s a graph showing the appearance and disappearance of Betti in different ranges of p:

This kind of “higher Erdös-Rényi theorem” is, to me, quite dramatic and unexpected.  (One consequence that I like a lot; if you condition on the complex having dimension d, i.e. d being the size of the largest clique in G(n,p), then with probability 1 the homology of the complex is supported in middle degree, just as you might want!)  And there’s other stuff there too — like a threshold for the fundamental group of X(n,p) to have property T.

For yet more about this area, see Kahle’s recent survey on the topology of random simplicial complexes.  The probability that a random graph has a spectral gap, the distribution of Betti numbers of X(n,p) in the regime where they’re nonzero, the behavior of torsion, etc., etc……