We start by reviewing results about Betti numbers of hyperkähler varieties.
The formulas are in a combinatorial form, which makes computation of the Betti
numbers for large examples feasible. We will consider limiting behaviours of the
graph of Betti numbers of such large hyperkähler manifolds. We find that in various
examples the Gauss, Gumbel and Airy distributions appear as the limit. This is joint
work (arXiv:1309.4914) with Fernando Rodriguez Villegas.