Abstract
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.
Keywords: Random walk; generalized lamplighter walk; wreath product; mixing time; relaxation
time.
Original language | English |
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Pages (from-to) | 1-23 |
Number of pages | 23 |
Journal | Electronic Journal of Probability |
Volume | 18 |
Issue number | 71 |
DOIs | |
Publication status | Published - 2013 |
Externally published | Yes |