14,886 research outputs found
Random walks on supercritical percolation clusters
We obtain Gaussian upper and lower bounds on the transition density q_t(x,y)
of the continuous time simple random walk on a supercritical percolation
cluster C_{\infty} in the Euclidean lattice. The bounds, analogous to Aronsen's
bounds for uniformly elliptic divergence form diffusions, hold with constants
c_i depending only on p (the percolation probability) and d. The irregular
nature of the medium means that the bound for q_t(x,\cdot) holds only for t\ge
S_x(\omega), where the constant S_x(\omega) depends on the percolation
configuration \omega.Comment: Published at http://dx.doi.org/10.1214/009117904000000748 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Restoration of isotropy on fractals
We report a new type of restoration of macroscopic isotropy (homogenization)
in fractals with microscopic anisotropy. The phenomenon is observed in various
physical setups, including diffusions, random walks, resistor networks, and
Gaussian field theories. The mechanism is unique in that it is absent in spaces
with translational invariance, while universal in that it is observed in a wide
class of fractals.Comment: 11 pages, REVTEX, 3 postscript figures. (Compressed and encoded
figures archived by "figure" command). To appear in Physical Review Letter
Exponential tail bounds for loop-erased random walk in two dimensions
Let be the number of steps of the loop-erasure of a simple random walk
on from the origin to the circle of radius . We relate the
moments of to , the probability that a random walk and an
independent loop-erased random walk both started at the origin do not intersect
up to leaving the ball of radius . This allows us to show that there exists
such that for all and all and hence to establish exponential moment bounds for
. This implies that there exists such that for all and all
,
Using similar techniques, we then establish a second moment result for a
specific conditioned random walk which enables us to prove that for any
such that for all and ,
Comment: Published in at http://dx.doi.org/10.1214/10-AOP539 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
The random conductance model with Cauchy tails
We consider a random walk in an i.i.d. Cauchy-tailed conductances
environment. We obtain a quenched functional CLT for the suitably rescaled
random walk, and, as a key step in the arguments, we improve the local limit
theorem for in [Ann. Probab. (2009). To appear],
Theorem 5.14, to a result which gives uniform convergence for
for all in a ball.Comment: Published in at http://dx.doi.org/10.1214/09-AAP638 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Energy inequalities for cutoff functions and some applications
We consider a metric measure space with a local regular Dirichlet form. We
establish necessary and sufficient conditions for upper heat kernel bounds with
sub-diffusive space-time exponent to hold. This characterization is stable
under rough isometries, that is it is preserved under bounded perturbations of
the Dirichlet form. Further, we give a criterion for stochastic completeness in
terms of a Sobolev inequality for cutoff functions. As an example we show that
this criterion applies to an anomalous diffusion on a geodesically incomplete
fractal space, where the well-established criterion in terms of volume growth
fails
The evolution of the cover time
The cover time of a graph is a celebrated example of a parameter that is easy
to approximate using a randomized algorithm, but for which no constant factor
deterministic polynomial time approximation is known. A breakthrough due to
Kahn, Kim, Lovasz and Vu yielded a (log log n)^2 polynomial time approximation.
We refine this upper bound, and show that the resulting bound is sharp and
explicitly computable in random graphs. Cooper and Frieze showed that the cover
time of the largest component of the Erdos-Renyi random graph G(n,c/n) in the
supercritical regime with c>1 fixed, is asymptotic to f(c) n \log^2 n, where
f(c) tends to 1 as c tends to 1. However, our new bound implies that the cover
time for the critical Erdos-Renyi random graph G(n,1/n) has order n, and shows
how the cover time evolves from the critical window to the supercritical phase.
Our general estimate also yields the order of the cover time for a variety of
other concrete graphs, including critical percolation clusters on the Hamming
hypercube {0,1}^n, on high-girth expanders, and on tori Z_n^d for fixed large
d. For the graphs we consider, our results show that the blanket time,
introduced by Winkler and Zuckerman, is within a constant factor of the cover
time. Finally, we prove that for any connected graph, adding an edge can
increase the cover time by at most a factor of 4.Comment: 14 pages, to appear in CP
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