  Bounds for the expected return probability of the delayed random walk on
finite clusters of an invariant percolation on transitive unimodular graphs are
derived. They are particularly suited for the case of critical Bernoulli
percolation and the associated heavy-tailed cluster size distributions. The
upper bound relies on the fact that cartesian products of finite graphs with
cycles of a certain minimal size are Hamiltonian. For critical Bernoulli bond
percolation on the homogeneous tree this bound is sharp. The asymptotic type of
the expected return probability for large times t in this case is of order of
the 3/4'th power of 1/t.
