Revelle, David Heat kernel asymptotics on the lamplighter group. (English) Zbl 1061.60112 Electron. Commun. Probab. 8, 142-154 (2003). 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 \(\alpha\). Cited in 18 Documents 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) Keywords:random walk; off-diagonal bound; heat kernel; trapping PDFBibTeX XMLCite \textit{D. Revelle}, Electron. Commun. Probab. 8, 142--154 (2003; Zbl 1061.60112) Full Text: DOI EuDML Online Encyclopedia of Integer Sequences: Cogrowth sequence of the lamplighter group Z_2 wr Z where wr denotes the wreath product.