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Heat kernel asymptotics on the lamplighter group. (English) Zbl 1061.60112

Summary: We show that, for one generating set, the on-diagonal decay of the heat kernel on the lamplighter group is asymptotic to \(c_1 n^{1/6}\exp[-c_2 n^{1/3}]\). We also make off-diagonal estimates which show that there is a sharp threshold for which elements have transition probabilities that are comparable to the return probability. The off-diagonal estimates also give an upper bound for the heat kernel that is uniformly summable in time. The methods used also apply to a one-dimensional trapping problem, and we compute the distribution of the walk conditioned on survival as well as a corrected asymptotic for the survival probability. Conditioned on survival, the position of the walker is shown to be concentrated within \(\alpha n^{1/3}\) of the origin for a suitable \(\alpha\).

MSC:

60K40 Other physical applications of random processes
60G50 Sums of independent random variables; random walks
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
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