  Suppose that G and H are finite, connected graphs, G regular, X is a lazy
random walk on G and Z is a reversible ergodic Markov chain on H. The
generalized lamplighter chain X* associated with X and Z is the random walk on
the wreath product H\wr G, the graph whose vertices consist of pairs (f,x)
where f=(f_v)_{v\in V(G)} is a labeling of the vertices of G by elements of H
and x is a vertex in G. In each step, X* moves from a configuration (f,x) by
updating x to y using the transition rule of X and then independently updating
both f_x and f_y according to the transition probabilities on H; f_z for z
different of x,y remains unchanged. We estimate the mixing time of X* in terms
of the parameters of H and G. Further, we show that the relaxation time of X*
is the same order as the maximal expected hitting time of G plus |G| times the
relaxation time of the chain on H.
