The Russo-Seymour-Welsh Theorem and the Equality of Critical Densities and the "Dual" Critical Densities for Continuum Percolation on $|mathbb{R}^2$

Roy, Rahul

doi:10.1214/aop/1176990632

Cited by 54 publications

(40 citation statements)

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“…A milestone within percolation theory was reached in 1980 with Kesten's proof that the thresholds for the existence of an infinite primal and an infinite dual component coincide [5]. Analogous results have since been obtained for models of percolation in the continuum of R 2 , such as Poisson Boolean percolation with bounded radii [8] and Poisson Voronoi percolation [2]. In our previous work [1], we characterized the phase transition of the 2 dimensional Poisson Boolean model in terms of crossing probabilities.…”

Section: Introductionmentioning

confidence: 90%

Existence of an unbounded vacant set for subcritical continuum percolation

Ahlberg¹,

Tassion²,

Teixeira³

2018

Electron. Commun. Probab.

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We consider the Poisson Boolean percolation model in R 2 , where the radii of each ball is independently chosen according to some probability measure with finite second moment. For this model, we show that the two thresholds, for the existence of an unbounded occupied and an unbounded vacant component, coincide. This complements a recent study of the sharpness of the phase transition in Poisson Boolean percolation by the same authors. As a corollary it follows that for Poisson Boolean percolation in R d , for any d ≥ 2, finite moment of order d is both necessary and sufficient for the existence of a nontrivial phase transition for the vacant set.

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Section: Introductionmentioning

confidence: 90%

Existence of an unbounded vacant set for subcritical continuum percolation

Ahlberg¹,

Tassion²,

Teixeira³

2018

Electron. Commun. Probab.

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“…continuum percolation) can be defined by taking a Poisson set of points W C ]R2 of intensity I and letting W be the set of points in the plane at distance at most r from W. Here, r is the parameter of the modeL (Alternatively, one may fix r = 1, say, and let the intensity of the Poisson process be the parameter, but this is essentially the same, by scaling.) The RussoSeymour-Welsh theorem is known for this model [1], [77], but the critical value of the parameter has not been identified. A nice feature which the model shares with Voronoi percolation is invariance under rotations.…”

Section: Thus It Is Natural To Askmentioning

confidence: 97%

Conformally invariant scaling limits: an overview and a collection of problems

Schramm

2011

Selected Works of Oded Schramm

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Abstract. Many mathematical models of statistical physics in two dimensions are either known or conjectured to exhibit conformal invariance. Over the years, physicists proposed predictions of various exponents describing the behavior of these models. Only recently have some of these predictions become accessible to mathematical proof. One of the new developments is the discovery of a one-parameter family of random curves called stochastic Loewner evolution or SLE. The SLE curves appear as limits of interfaces or paths occurring in a variety of statistical physics models as the mesh of the grid on which the model is defined tends to zero.The main purpose of this article is to list a collection of open problems. Some of the open problems indicate aspects of the physics knowledge that have not yet been understood mathematically. Other problems are questions about the nature of the SLE curves themselves. Before we present the open problems, the definition of SLE will be motivated and explained, and a brief sketch of recent results will be presented. Mathematics Subject Classification (2000). Primary 60K35; Secondary 82B20, 82B43, 30C35.

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“…Finally, for a fixed L > 2R, as A 1" oo, So, by Harris' FKG inequality (see Roy [11] for a generalization which allows us to apply the inequality in our model), (a), (b) and (c) along with the preceding discussion, imply that for A > A0, some point in S(0, 0) is in an infinite occupied cluster with probability at least 1/2. Since, with positive probability, we can connect a ray in S(0, 0) to the ray ~0, we have, for ), > A0, ~0 is in an infinite occupied cluster with positive probability, i.e., A ~ A0 < oe.…”

Section: )~C < C~mentioning

confidence: 70%

Critical phenomenon for a percolation model

Roy

1992

Acta Appl Math

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We consider a percolation model which consists of oriented lines placed randomly on the plane. The lines are of random length and at a random angle with respect to the horizontal axis and are placed according to a Poisson point process; the length, angle, and orientation being independent of the underlying Poisson process. We establish a critical behaviour of this model, i.e., percolation occurs for large intensity of the Poisson process and does not occur for smaller intensities. In the special case when the lines are of fixed unit length and are either oriented vertically up or oriented horizontally to the left, with probability p or (l-p), respectively, we obtain a lower bound on the critical intensity of percolation.Mathematics Subject Classifications (1991), 60K35, 82B43.

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The Russo-Seymour-Welsh Theorem and the Equality of Critical Densities and the "Dual" Critical Densities for Continuum Percolation on $|mathbb{R}^2$

Cited by 54 publications

References 0 publications

Existence of an unbounded vacant set for subcritical continuum percolation

Existence of an unbounded vacant set for subcritical continuum percolation

Conformally invariant scaling limits: an overview and a collection of problems

Critical phenomenon for a percolation model

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