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

Bounds for the annealed return probability on large finite percolation clusters