256 research outputs found

    Percolation on dual lattices with k-fold symmetry

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    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

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    The Burton--Keane theorem for the almost-sure uniqueness of infinite clusters is a landmark of stochastic geometry. Let μ\mu be a translation-invariant probability measure with the finite-energy property on the edge-set of a dd-dimensional lattice. The theorem states that the number II of infinite components satisfies μ(I∈{0,1})=1\mu(I\in\{0,1\})=1. 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

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    Given a graph GG, we consider the model where GG is given a random orientation by giving each edge a random direction. It is proven that for a,b,s∈V(G)a,b,s\in V(G), the events {s→a}\{s\to a\} and {s→b}\{s\to b\} are positively correlated. This correlation persists, perhaps unexpectedly, also if we first condition on \{s\nto t\} for any vertex t≠st\neq s. With this conditioning it is also true that {s→b}\{s\to b\} and {a→t}\{a\to t\} 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

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    For ordinary (independent) percolation on a large class of lattices it is well known that below the critical percolation parameter pcp_c the cluster size distribution has exponential decay and that power-law behavior of this distribution can only occur at pcp_c. 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'') 2D2D contact process with parameter λ\lambda. We show, using techniques from Bollob\'{a}s and Riordan [8, 11], that for the upper invariant measure νˉλ{\bar{\nu}}_{\lambda} of this process the percolation transition is sharp. If λ\lambda is such that (νˉλ{\bar{\nu}}_{\lambda}-a.s.) there are no infinite clusters, then for all parameter values below λ\lambda 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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