  Let $G$ be a finitely generated group and $X$ its Cayley graph with respect
to a finite, symmetric generating set $S$. Furthermore, let $H$ be a finite
group and $H \wr G$ the lamplighter group (wreath product) over $G$ with group
of "lamps" $H$. We show that the spectral measure (Plancherel measure) of any
symmetric "switch--walk--switch" random walk on $H \wr G$ coincides with the
expected spectral measure (integrated density of states) of the random walk
with absorbing boundary on the cluster of the group identity for Bernoulli site
percolation on $X$ with parameter $p = 1/|H|$. The return probabilities of the
lamplighter random walk coincide with the expected (annealed) return
probabilites on the percolation cluster. In particular, if the clusters of
percolation with parameter $p$ are almost surely finite then the spectrum of
the lamplighter group is pure point. This generalizes results of Grigorchuk and
Zuk, resp. Dicks and Schick regarding the case when $G$ is infinite cyclic.
Analogous results relate bond percolation with another lamplighter random walk.
In general, the integrated density of states of site (or bond) percolation with
arbitrary parameter $p$ is always related with the Plancherel measure of a
convolution operator by a signed measure on $H \wr G$, where $H = Z$ or another
suitable group.
