809 research outputs found
Configurations of balls in Euclidean space that Brownian motion cannot avoid
We consider a collection of balls in Euclidean space and the problem of
determining if Brownian motion has a positive probability of avoiding all the
ball
On the critical parameter of interlacement percolation in high dimension
The vacant set of random interlacements on , , has
nontrivial percolative properties. It is known from Sznitman [Ann. Math. 171
(2010) 2039--2087], Sidoravicius and Sznitman [Comm. Pure Appl. Math. 62 (2009)
831--858] that there is a nondegenerate critical value such that the
vacant set at level percolates when and does not percolate when
. We derive here an asymptotic upper bound on , as goes to
infinity, which complements the lower bound from Sznitman [Probab. Theory
Related Fields, to appear]. Our main result shows that is equivalent to
for large and thus has the same principal asymptotic behavior as
the critical parameter attached to random interlacements on -regular trees,
which has been explicitly computed in Teixeira [Electron. J. Probab. 14 (2009)
1604--1627].Comment: Published in at http://dx.doi.org/10.1214/10-AOP545 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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