2,011 research outputs found
On monochromatic arm exponents for 2D critical percolation
We investigate the so-called monochromatic arm exponents for critical
percolation in two dimensions. These exponents, describing the probability of
observing j disjoint macroscopic paths, are shown to exist and to form a
different family from the (now well understood) polychromatic exponents. More
specifically, our main result is that the monochromatic j-arm exponent is
strictly between the polychromatic j-arm and (j+1)-arm exponents.Comment: Published in at http://dx.doi.org/10.1214/10-AOP581 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Near-critical percolation with heavy-tailed impurities, forest fires and frozen percolation
Consider critical site percolation on a "nice" planar lattice: each vertex is
occupied with probability , and vacant with probability .
Now, suppose that additional vacancies ("holes", or "impurities") are created,
independently, with some small probability, i.e. the parameter is
replaced by , for some small . A celebrated
result by Kesten says, informally speaking, that on scales below the
characteristic length , the connection probabilities
remain of the same order as before. We prove a substantial and subtle
generalization to the case where the impurities are not only microscopic, but
allowed to be "mesoscopic".
This generalization, which is also interesting in itself, was motivated by
our study of models of forest fires (or epidemics). In these models, all
vertices are initially vacant, and then become occupied at rate . If an
occupied vertex is hit by lightning, which occurs at a (typically very small)
rate , its entire occupied cluster burns immediately, so that all its
vertices become vacant.
Our results for percolation with impurities turn out to be crucial for
analyzing the behavior of these forest fire models near and beyond the critical
time (i.e. the time after which, in a forest without fires, an infinite cluster
of trees emerges). In particular, we prove (so far, for the case when burnt
trees do not recover) the existence of a sequence of "exceptional scales"
(functions of ). For forests on boxes with such side lengths, the impact
of fires does not vanish in the limit as .Comment: 67 pages, 15 figures (some small corrections and improvements, one
additional figure); version to be submitte
Two-dimensional volume-frozen percolation: exceptional scales
We study a percolation model on the square lattice, where clusters "freeze"
(stop growing) as soon as their volume (i.e. the number of sites they contain)
gets larger than N, the parameter of the model. A model where clusters freeze
when they reach diameter at least N was studied in earlier papers. Using volume
as a way to measure the size of a cluster - instead of diameter - leads, for
large N, to a quite different behavior (contrary to what happens on the binary
tree, where the volume model and the diameter model are "asymptotically the
same"). In particular, we show the existence of a sequence of "exceptional"
length scales.Comment: 20 pages, 2 figure
Multiples bris communs en variance et en moyenne des panels de séries temporelles
Rapport de recherche présenté à la Faculté des arts et des sciences en vue de l'obtention du grade de Maîtrise en sciences économiques
A percolation process on the binary tree where large finite clusters are frozen
We study a percolation process on the planted binary tree, where clusters
freeze as soon as they become larger than some fixed parameter N. We show that
as N goes to infinity, the process converges in some sense to the frozen
percolation process introduced by Aldous. In particular, our results show that
the asymptotic behaviour differs substantially from that on the square lattice,
on which a similar process has been studied recently by van den Berg, de Lima
and Nolin.Comment: 11 page
Percolation on uniform infinite planar maps
We construct the uniform infinite planar map (UIPM), obtained as the n \to
\infty local limit of planar maps with n edges, chosen uniformly at random. We
then describe how the UIPM can be sampled using a "peeling" process, in a
similar way as for uniform triangulations. This process allows us to prove that
for bond and site percolation on the UIPM, the percolation thresholds are
p_c^bond=1/2 and p_c^site=2/3 respectively. This method also works for other
classes of random infinite planar maps, and we show in particular that for bond
percolation on the uniform infinite planar quadrangulation, the percolation
threshold is p_c^bond=1/3.Comment: 26 pages, 9 figure
- …