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1.
Let be a finitely generated group and X its Cayley graph with respect to a finite, symmetric generating set S. Furthermore, let be a finite group and the lamplighter group (wreath product) over with group of “lamps” . We show that the spectral measure (Plancherel measure) of any symmetric “switch–walk–switch” random walk on coincides with the expected spectral measure (integrated density of states) of the random walk with absorbing boundary on
the cluster of the group identity for Bernoulli site percolation on X with parameter . The return probabilities of the lamplighter random walk coincide with the expected (annealed) return probabilities on the
percolation cluster. In particular, if the clusters of percolation with parameter are almost surely finite then the spectrum of the lamplighter group is pure point. This generalizes results of Grigorchuk
and Żuk, resp. Dicks and Schick regarding the case when is infinite cyclic. Analogous results relate bond percolation with another lamplighter random walk. In general, the integrated
density of states of site (or bond) percolation with arbitrary parameter is always related with the Plancherel measure of a convolution operator by a signed measure on , where or another suitable group.
M. Neuhauser’s research supported by the Marie-Curie Excellence Grant MEXT-CT-2004-517154.
The research of W. Woess was partially supported by Austrian Science Fund (FWF) P18703-N18. 相似文献
2.
Summary. Consider (independent) first-passage percolation on the edges of ℤ
2
. Denote the passage time of the edge e in ℤ
2
by t(e), and assume that P{t(e) = 0} = 1/2, P{0<t(e)<C
0
} = 0 for some constant C
0
>0 and that E[t
δ
(e)]<∞ for some δ>4. Denote by b
0,n
the passage time from 0 to the halfplane {(x,y): x ≧ n}, and by T(
0
,nu) the passage time from 0 to the nearest lattice point to nu, for u a unit vector. We prove that there exist constants 0<C
1
, C
2
<∞ and γ
n
such that C
1
(
log
n)
1/2
≦γ
n
≦ C
2
(
log
n)
1/2
and such that γ
n
−1
[b
0,n
−Eb
0,n
] and (√ 2γ
n
)
−1
[T(
0
,nu) − ET(
0
,nu)] converge in distribution to a standard normal variable (as n →∞, u fixed).
A similar result holds for the site version of first-passage percolation on ℤ
2
, when the common distribution of the passage times {t(v)} of the vertices satisfies P{t(v) = 0} = 1−P{t(v) ≧ C
0
} = p
c
(ℤ
2
,
site
) := critical probability of site percolation on ℤ
2
, and E[t
δ
(u)]<∞ for some δ>4.
Received: 6 February 1996 / In revised form: 17 July 1996 相似文献
3.
The authors localize the blow-up points of positive solutions of the systemu
t
=Δu,v
t
=Δv with conditions
at the boundary of a bounded smooth domain Θ under some restrictions off andg and the initial data (Δu
0, Δν0>c>0).
If Θ is a ball, the hypothesis on the initial data can be removed.
Supported by Universidad de Buenos Aires under grant EX071 and CONICET. 相似文献
4.
The authors rigorously prove that the exponent for the mean square displacement of self-avoiding random walk on the Sierpinski gasket is
相似文献
5.
T. S. Mountford 《Probability Theory and Related Fields》1992,93(1):67-76
We consider the nearest particle system which gives birth rate to each vacant interval, concentrated on the interval's midpoint(s). We prove that a critical value for exists and equals one. The proof extends to a large class of nearest particle systems. This paper solves a problem suggested by Liggett (1985).In the following we deal with nearest particle systems {
t
:t0}. These can be described as particle systems with the following flip rates:
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