## Electronic Communications in Probability

### Heat Kernel Asymptotics on the Lamplighter Group

David Revelle

#### Abstract

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 $\alpha$.

#### Article information

Source
Electron. Commun. Probab., Volume 8 (2003), paper no. 16, 142-154.

Dates
Accepted: 10 November 2003
First available in Project Euclid: 18 May 2016

https://projecteuclid.org/euclid.ecp/1463608900

Digital Object Identifier
doi:10.1214/ECP.v8-1092

Mathematical Reviews number (MathSciNet)
MR2042753

Zentralblatt MATH identifier
1061.60112

Rights

#### Citation

Revelle, David. Heat Kernel Asymptotics on the Lamplighter Group. Electron. Commun. Probab. 8 (2003), paper no. 16, 142--154. doi:10.1214/ECP.v8-1092. https://projecteuclid.org/euclid.ecp/1463608900

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