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Heat Kernel Asymptotics on the Lamplighter Group
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David Revelle, UC Berkeley
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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 $alpha$.
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Full text: PDF
Pages: 142-154
Published on: November 10, 2003
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Electronic Communications in Probability. ISSN: 1083-589X |
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