256 research outputs found
Percolation on dual lattices with k-fold symmetry
Zhang found a simple, elegant argument deducing the non-existence of an
infinite open cluster in certain lattice percolation models (for example, p=1/2
bond percolation on the square lattice) from general results on the uniqueness
of an infinite open cluster when it exists; this argument requires some
symmetry. Here we show that a simple modification of Zhang's argument requires
only 2-fold (or 3-fold) symmetry, proving that the critical probabilities for
percolation on dual planar lattices with such symmetry sum to 1. Like Zhang's
argument, our extension applies in many contexts; in particular, it enables us
to answer a question of Grimmett concerning the anisotropic random cluster
model on the triangular lattice.Comment: 11 pages, 1 figure. Revised with applications added; to appear in
Random Structures and Algorithm
Uniqueness and multiplicity of infinite clusters
The Burton--Keane theorem for the almost-sure uniqueness of infinite clusters
is a landmark of stochastic geometry. Let be a translation-invariant
probability measure with the finite-energy property on the edge-set of a
-dimensional lattice. The theorem states that the number of infinite
components satisfies . The proof is an elegant and
minimalist combination of zero--one arguments in the presence of amenability.
The method may be extended (not without difficulty) to other problems including
rigidity and entanglement percolation, as well as to the Gibbs theory of
random-cluster measures, and to the central limit theorem for random walks in
random reflecting labyrinths. It is a key assumption on the underlying graph
that the boundary/volume ratio tends to zero for large boxes, and the picture
for non-amenable graphs is quite different.Comment: Published at http://dx.doi.org/10.1214/074921706000000040 in the IMS
Lecture Notes--Monograph Series
(http://www.imstat.org/publications/lecnotes.htm) by the Institute of
Mathematical Statistics (http://www.imstat.org
A note on correlations in randomly oriented graphs
Given a graph , we consider the model where is given a random
orientation by giving each edge a random direction. It is proven that for
, the events and are positively
correlated. This correlation persists, perhaps unexpectedly, also if we first
condition on \{s\nto t\} for any vertex . With this conditioning it
is also true that and are negatively correlated.
A concept of increasing events in random orientations is defined and a
general inequality corresponding to Harris inequality is given.
The results are obtained by combining a very useful lemma by Colin McDiarmid
which relates random orientations with edge percolation, with results by van
den Berg, H\"aggstr\"om, Kahn on correlation inequalities for edge percolation.
The results are true also for another model of randomly directed graphs.Comment: 7 pages. The main lemma was first published by Colin McDiarmid.
Relevant reference added and text rewritten to reflect this fac
Sharpness of the percolation transition in the two-dimensional contact process
For ordinary (independent) percolation on a large class of lattices it is
well known that below the critical percolation parameter the cluster size
distribution has exponential decay and that power-law behavior of this
distribution can only occur at . This behavior is often called ``sharpness
of the percolation transition.'' For theoretical reasons, as well as motivated
by applied research, there is an increasing interest in percolation models with
(weak) dependencies. For instance, biologists and agricultural researchers have
used (stationary distributions of) certain two-dimensional contact-like
processes to model vegetation patterns in an arid landscape (see [20]). In that
context occupied clusters are interpreted as patches of vegetation. For some of
these models it is reported in [20] that computer simulations indicate
power-law behavior in some interval of positive length of a model parameter.
This would mean that in these models the percolation transition is not sharp.
This motivated us to investigate similar questions for the ordinary (``basic'')
contact process with parameter . We show, using techniques from
Bollob\'{a}s and Riordan [8, 11], that for the upper invariant measure
of this process the percolation transition is sharp. If
is such that (-a.s.) there are no infinite
clusters, then for all parameter values below the cluster-size
distribution has exponential decay.Comment: Published in at http://dx.doi.org/10.1214/10-AAP702 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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