  A comparison technique for finite random walks on finite graphs is
introduced, using the well-known interlacing method. It yields improved return
probability bounds. A key feature is the incorporation of parts of the spectrum
of the transition matrix other than just the principal eigenvalue. As an
application, an upper bound of the expected return probability of a random walk
with symmetric transition probabilities is found. In this case, the state space
is a random partial graph of a regular graph of bounded geometry and transitive
automorphism group. The law of the random edge-set is assumed to be stationary
with respect to some transitive subgroup of the automorphism group (`invariant
percolation'). Given that this subgroup is unimodular, it is shown that
stationarity strengthens the upper bound of the expected return probability,
compared with standard bounds derived from the Cheeger inequality.
