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Lima, Bernardo N. B. de; Sanchis, Rémy; Silva, Roger W. C.

doi:10.1016/j.spa.2011.05.009

Cited by 6 publications

(3 citation statements)

References 4 publications

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“…Consider the graph having Z as vertex set and all edges of the form {x, x ± e i } and {x, x ± k ⋅ e i } for some k ⩾ 2. It was shown in [9] that the critical probability for Bernoulli bond percolation on this graph converges to that of Z 2 as k → ∞. This result, later generalized in [15], is a particular instance of Schramm's conjecture [7] that the percolation threshold for transitive graphs is a local property.…”

Section: Introductionmentioning

confidence: 77%

“…For vertices u ′ , v ′ with v ′ ∈ prog(u ′ ), let dist(u ′ , v ′ ) denote the length of the unique path of short edges from u ′ to v ′ . Then, (9)…”

Section: Comparison Of Different Rangesmentioning

confidence: 99%

“…This result, later generalized in [15], is a particular instance of Schramm's conjecture [7] that the percolation threshold for transitive graphs is a local property. The convergence proved in [9] is conjectured to be monotone, that is, the percolation threshold for the above graph should be decreasing in the length k of long edges. 1 1 Phase space and critical curve separating the percolative region  k from the non-percolative region  k .…”

Section: Introductionmentioning

confidence: 97%

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Monotonicity and phase diagram for multirange percolation on oriented trees

Lima

Rolla

Valesin

2018

Random Struct Algorithms

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We consider Bernoulli bond percolation on oriented regular trees, where besides the usual short bonds, all bonds of a certain length are added. Independently, short bonds are open with probability p and long bonds are open with probability q. We study properties of the critical curve which delimits the set of pairs (p,q) for which there are almost surely no infinite paths. We also show that this curve decreases with respect to the length of the long bonds.

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Section: Introductionmentioning

confidence: 77%

“…For vertices u ′ , v ′ with v ′ ∈ prog(u ′ ), let dist(u ′ , v ′ ) denote the length of the unique path of short edges from u ′ to v ′ . Then, (9)…”

Section: Comparison Of Different Rangesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 97%

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Monotonicity and phase diagram for multirange percolation on oriented trees

Lima

Rolla

Valesin

2018

Random Struct Algorithms

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Multirange percolation on oriented trees: Critical curve and limit behavior

Lima

Szabó

Valesin

2022

Random Struct Algorithms

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We consider an inhomogeneous oriented percolation model introduced by de Lima, Rolla and Valesin. In this model, the underlying graph is an oriented rooted tree in which each vertex points to each of its 𝑑 children with "short" edges, and in addition, each vertex points to each of its 𝑑 k descendant at a fixed distance k with "long" edges. A bond percolation process is then considered on this graph, with the prescription that independently, short edges are open with probability p and long edges are open with probability q. We study the behavior of the critical curve q c = q c (p, k, 𝑑): we find the first two terms in the expansion of q c as k → ∞. We also prove limit theorems for the percolation cluster in the supercritical, subcritical and critical regimes.

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A Note on Schramm’s Locality Conjecture for Random-Cluster Models

Duminil-Copin

Tassion

2019

Sojourns in Probability Theory and Statistical Physics - II

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In this note, we discuss a generalization of Schramm's locality conjecture to the case of random-cluster models. We give some partial (modest) answers, and present several related open questions. Our main result is to show that the critical inverse temperature of the Potts model on Z r × (Z 2nZ) d−r (with r ≥ 3) converges to the critical inverse temperature of the model on Z d as n tends to infinity. Our proof relies on the infrared bound and, contrary to the corresponding statement for Bernoulli percolation, does not involve renormalization arguments.

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Critical point and percolation probability in a long range site percolation model on Zd

Cited by 6 publications

References 4 publications

Monotonicity and phase diagram for multirange percolation on oriented trees

Monotonicity and phase diagram for multirange percolation on oriented trees

Multirange percolation on oriented trees: Critical curve and limit behavior

A Note on Schramm’s Locality Conjecture for Random-Cluster Models

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