  We study the topology of a random cubical complex associated to Bernoulli
site percolation on a cubical grid. We begin by establishing a limit law for
homotopy types. More precisely, looking within an expanding window, we define a
sequence of normalized counting measures (counting connected components
according to homotopy type), and we show that this sequence of random
probability measures converges in probability to a deterministic probability
measure. We then investigate the dependence of the limiting homotopy measure on
the coloring probability $p$, and our results show a qualitative change in the
homotopy measure as $p$ crosses the percolation threshold $p=p_c$. Specializing
to the case of $d=2$ dimensions, we also present empirical results that raise
further questions on the $p$-dependence of the limiting homotopy measure.
