RATES OF CONVERGENCE FOR LAMPLIGHTER PROCESSES

Consider a graph, G, for which the vertices can have two modes, 0 or 1 . Suppose that a particle moves around on G according to a discrete ti me Markov chain with the following rules. With (strictly positive) pro babilities p(m), p(c) and p(r) it moves to a randomly chosen neighbour , changes the mode of the vertex it is at or just stands still, respec tively. We call such a random process a (p(m), p(c), p(r))-lamplighter process on G. Assume that the process starts with the particle in a f ixed position and with all vertices having mode 0. The convergence rat e to stationarity in terms of the total variation norm is studied for the special cases with G = H-N, the complete graph with N vertices, an d when G = Z mod N. In the former case we prove that as N --> infinity , ((2p(c) + p(m))/4p(c)p(m))N log N is a threshold for the convergence rate. In the latter case we show that the convergence rate is asympto tically determined by the cover time C-N in that the total variation n orm after aN(2) steps is given by P(C-N > aN(2)). The limit of this pr obability can in turn be calculated by considering a Brownian motion w ith two absorbing barriers. In particular, this means that there is no threshold for this case.