508 research outputs found
The expected number of critical percolation clusters intersecting a line segment
We study critical percolation on a regular planar lattice. Let be
the expected number of open clusters intersecting or hitting the line segment
. (For the subscript we either take , when we restrict
to the upper halfplane, or , when we consider the full lattice).
Cardy (2001) (see also Yu, Saleur and Haas (2008)) derived heuristically that
, where
is some constant. Recently Kov\'{a}cs, Igl\'{o}i and Cardy (2012) derived
heuristically (as a special case of a more general formula) that a similar
result holds for with the constant
replaced by . In this paper we give, for site
percolation on the triangular lattice, a rigorous proof for the formula of
above, and a rigorous upper bound for the prefactor of the
logarithm in the formula of .Comment: Final version, appeared in Elect.Comm.Probab. 21 (2016
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
Sublinearity of the travel-time variance for dependent first-passage percolation
Let be the set of edges of the -dimensional cubic lattice
, with , and let , be nonnegative values.
The passage time from a vertex to a vertex is defined as
, where the infimum is over all
paths from to , and the sum is over all edges of .
Benjamini, Kalai and Schramm [2] proved that if the 's are i.i.d.
two-valued positive random variables, the variance of the passage time from the
vertex 0 to a vertex is sublinear in the distance from 0 to . This
result was extended to a large class of independent, continuously distributed
-variables by Bena\"{\i}m and Rossignol [1]. We extend the result by
Benjamini, Kalai and Schramm in a very different direction, namely to a large
class of models where the 's are dependent. This class includes, among
other interesting cases, a model studied by Higuchi and Zhang [9], where the
passage time corresponds with the minimal number of sign changes in a
subcritical "Ising landscape."Comment: Published in at http://dx.doi.org/10.1214/10-AOP631 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
The gaps between the sizes of large clusters in 2D critical percolation
Consider critical bond percolation on a large 2n by 2n box on the square
lattice. It is well-known that the size (i.e. number of vertices) of the
largest open cluster is, with high probability, of order n^2 \pi(n), where
\pi(n) denotes the probability that there is an open path from the center to
the boundary of the box. The same result holds for the second-largest cluster,
the third largest cluster etcetera. Jarai showed that the differences between
the sizes of these clusters is, with high probability, at least of order
\sqrt{n^2 \pi(n)}. Although this bound was enough for his applications (to
incipient infinite clusters), he believed, but had no proof, that the
differences are in fact of the same order as the cluster sizes themselves, i.e.
n^2 \pi(n). Our main result is a proof that this is indeed the case.Comment: 10 page
A signal-recovery system: asymptotic properties and construction of an infinite volume limit
We consider a linear sequence of `nodes', each of which can be in state 0
(`off') or 1 (`on'). Signals from outside are sent to the rightmost node and
travel instantaneously as far as possible to the left along nodes which are
`on'. These nodes are immediately switched off, and become on again after a
recovery time. The recovery times are independent exponentially distributed
random variables.
We present properties for finite systems and use some of these properties to
construct an infinite-volume extension, with signals `coming from infinity'.
This construction is related to a question by D. Aldous and we expect that it
sheds some light on, and stimulates further investigation of, that question.Comment: 16 page
On the size of the largest cluster in 2D critical percolation
We consider (near-)critical percolation on the square lattice. Let M_n be the
size of the largest open cluster contained in the box [-n,n]^2, and let pi(n)
be the probability that there is an open path from O to the boundary of the
box. It is well-known that for all 0< a < b the probability that M_n is smaller
than an^2 pi(n) and the probability that M_n is larger than bn^2 pi(n) are
bounded away from 0 as n tends to infinity. It is a natural question, which
arises for instance in the study of so-called frozen-percolation processes, if
a similar result holds for the probability that M_n is between an^2 pi(n) and
bn^2 pi(n). By a suitable partition of the box, and a careful construction
involving the building blocks, we show that the answer to this question is
affirmative. The `sublinearity' of 1/pi(n) appears to be essential for the
argument.Comment: 12 pages, 3 figures, minor change
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
An OSSS-type inequality for uniformly drawn subsets of fixed size
The OSSS inequality [O'Donnell, Saks, Schramm and Servedio, 46th Annual IEEE
Symposium on Foundations of Computer Science (FOCS'05), Pittsburgh (2005)]
gives an upper bound for the variance of a function f of independent 0-1 valued
random variables, in terms of the influences of these random variables and the
computational complexity of a (randomised) algorithm for determining the value
of f. Duminil-Copin, Raoufi and Tassion [Annals of Mathematics 189, 75-99
(2019)] obtained a generalization to monotonic measures and used it to prove
new results for Potts models and random-cluster models. Their generalization of
the OSSS inequality raises the question if there are still other measures for
which a version of that inequality holds. We derive a version of the OSSS
inequality for a family of measures that are far from monotonic, namely the
k-out-of-n measures (these measures correspond with drawing k elements from a
set of size n uniformly). We illustrate the inequality by studying the event
that there is an occupied horizontal crossing of an R times R box on the
triangular lattice in the site percolation model where exactly half of the
vertices in the box are occupied.Comment: 23 pages. Theorem 1.3 has been added. Section 5.2 discusses (ideas
for) other proofs of Theorem 5.2. References [7], [10] and [17] adde
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