Dynamical stability of percolation for some interacting particle systems and ɛ-movability

Broman, Erik I.; Steif, Jeffrey E.

doi:10.1214/009117905000000602

Cited by 12 publications

(20 citation statements)

References 23 publications

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“…Here, |z| denotes the distance to 0, which is the apex of the cone C θ . One can prove that 6) in the very same way that we have justified (7.1). By Theorem 8.1 (and easy algebraic manipulation), it therefore suffices to prove that…”

Section: Upper Bounds For K-arm Timessupporting

confidence: 75%

“…Next, analogous questions for the Boolean model where the points undergo independent Brownian motions was studied in [5]. Analogous questions for the lattice case for certain interacting particle systems (where updates are not done in an independent fashion) are studied in [6]. Finally in [3], it is shown that there are exceptional two dimensionl slices for the Boolean model in four dimensions.…”

Section: Theorem 13 Almost Surely the Set Of Times T ∈ [0 1] Suchmentioning

confidence: 99%

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Quantitative noise sensitivity and exceptional times for percolation

2010

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One goal of this paper is to prove that dynamical critical site percolation on the planar triangular lattice has exceptional times at which percolation occurs. In doing so, new quantitative noise sensitivity results for percolation are obtained. The latter is based on a novel method for controlling the "level k" Fourier coefficients via the construction of a randomized algorithm which looks at random bits, outputs the value of a particular function but looks at any fixed input bit with low probability. We also obtain upper and lower bounds on the Hausdorff dimension of the set of percolating times. We then study the problem of exceptional times for certain "k-arm" events on wedges and cones. As a corollary of this analysis, we prove, among other things, that there are no times at which there are two infinite "white" clusters, obtain an upper bound on the Hausdorff dimension of the set of times at which there are both an infinite white cluster and an infinite black cluster and prove that for dynamical critical bond percolation on the square grid there are no exceptional times at which 3 disjoint infinite clusters are present.

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Section: Upper Bounds For K-arm Timessupporting

confidence: 75%

Section: Theorem 13 Almost Surely the Set Of Times T ∈ [0 1] Suchmentioning

confidence: 99%

Quantitative noise sensitivity and exceptional times for percolation

2010

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“…6. The analogous state ments for critical percolati on in cones and wedges a lso holds, with similar proofs.…”

Section: )supporting

confidence: 57%

Quantitative noise sensitivity and exceptional times for percolation

Schramm

Steif

2011

Selected Works of Oded Schramm

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One goal of this paper is to prove that dy namical critical site percolation on the planar triangular lau ice has exceptional times at which percolation occurs. In doing so, new qllQmirarive noise sensitivity results for percolation are obtai ned. The latter is based on a novel method for controlling the "level k" Fouri er coeffici ents via the construction of a randomized algori thm which looks at random bits, outputs the value of a particular function but looks at any fi xed input bit with low probability. We also obtain upper and lower bounds on the Hausdorff di mension of the set of percolating times. We then study the problem of except ional times for certain "karm" events on wedges and cones. As a corollary of this analysis, we prove, among other things, that there are no ti mes at which there are two in fin ite "white" clusters, obtain an upper bound on the Hausdorff d imension of the set of times at which there are both an infini te wh ite cluster and an infinite black cluster and prove that for dynamical critical bond percolation on the square grid there are no exceptional times at which three disjoint infi nite clusters are present.

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“…where the inequality comes from the unconditioned version of (5). Taking N = T , we see that (14), (15) cannot hold for all N, T .…”

Section: The One Dimensional Contact Processmentioning

confidence: 95%

Stochastic domination: the contact process, Ising models and FKG measures☆

Liggett

Steif

2006

Annales de l'Institut Henri Poincare (B) Probability and Statis

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Abstract. We prove for the contact process on Z d , and many other graphs, that the upper invariant measure dominates a homogeneous product measure with large density if the infection rate λ is sufficiently large. As a consequence, this measure percolates if the corresponding product measure percolates. We raise the question of whether domination holds in the symmetric case for all infinite graphs of bounded degree. We study some asymmetric examples which we feel shed some light on this question. We next obtain necessary and sufficient conditions for domination of a product measure for "downward" FKG measures. As a consequence of this general result, we show that the plus and minus states for the Ising model on Z d dominate the same set of product measures. We show that this latter fact fails completely on the homogenous 3-ary tree. We also provide a different distinction between Z d and the homogenous 3-ary tree concerning stochastic domination and Ising models; while it is known that the plus states for different temperatures on Z d are never stochastically ordered, on the homogenous 3-ary tree, almost the complete opposite is the case. Next, we show that on Z d , the set of product measures which the plus state for the Ising model dominates is strictly increasing in the temperature. Finally, we obtain a necessary and sufficient condition for a finite number of variables, which are both FKG and exchangeable, to dominate a given product measure.

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Dynamical stability of percolation for some interacting particle systems and ɛ-movability

Cited by 12 publications

References 23 publications

Quantitative noise sensitivity and exceptional times for percolation

Quantitative noise sensitivity and exceptional times for percolation

Quantitative noise sensitivity and exceptional times for percolation

Stochastic domination: the contact process, Ising models and FKG measures☆

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