Mixing and relaxation time for random walk on wreath product graphs

J. Komjáthy, Y. Peres

Research output: Contribution to journalArticleAcademicpeer-review

1 Citation (Scopus)
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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 languageEnglish
Pages (from-to)1-23
Number of pages23
JournalElectronic Journal of Probability
Volume18
Issue number71
DOIs
Publication statusPublished - 2013
Externally publishedYes

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