  We study torsion in homology of the random $d$-complex $Y \sim Y_d(n,p)$
experimentally. Our experiments suggest that there is almost always a moment in
the process where there is an enormous burst of torsion in homology
$H_{d-1}(Y)$. This moment seems to coincide with the phase transition studied
in \cite{AL,LP,LP3} , where cycles in $H_d(Y)$ first appear with high
probability.
  Our main study is the limiting distribution on the $q$-part of the torsion
subgroup of $H_{d-1}(Y)$ for small primes $q$. We find strong evidence for a
limiting Cohen--Lenstra distribution, where the probability that the $q$-part
is isomorphic to a given $q$-group $H$ is inversely proportional to the order
of the automorphism group $|\mbox{Aut}(H)|$.
  We also study the torsion in homology of the uniform random $\Q$-acyclic
$2$-complex. This model is analogous to a uniform spanning tree on a complete
graph, but more complicated topologically since Kalai showed that the expected
order of the torsion group is exponentially large in $n^2$ \cite{Kalai}. We
give experimental evidence that in this model also, the torsion is
Cohen--Lenstra distributed in the limit.
