Sharp metastability threshold for an anisotropic bootstrap percolation model

Duminil-Copin, Hugo; Enter, Aernout van

doi:10.1214/11-aop722

Cited by 33 publications

(65 citation statements)

References 38 publications

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“…Note that in the case b = 2 this reduces to Theorem 1.1. We remark that Theorem 1.4 follows from a corresponding generalisation of Theorem 1.3, with the constants 1 6 and 1 3 log 8 3e replaced by 8 This is contrary to the claim in [23,Section 1]. See also [24], the erratum to [23].…”

Section: 5mentioning

confidence: 91%

“…The arguments developed in [23] can be applied to prove that the leading order behaviour of p c for the (…”

Section: 5mentioning

confidence: 99%

“…Combining the techniques of [23] with those introduced in this paper, it is possible to prove the following stronger bounds:…”

Section: 5mentioning

confidence: 99%

“…For filling a droplet, this means that one should find an optimal "growth trajectory": a sequence of dimensions from which a very small infected area (a "seed") steadily grows to fill up the entire droplet. For the anisotropic model, in [23], the first and second authors determined this trajectory to be close to x = e 3py 3p , where x and y denote the horizontal and vertical dimensions of the seed as it grows. This approximation was enough to yield the first term of p c .…”

Section: 7mentioning

confidence: 99%

“…In 2007, the second and third authors [26] determined the correct order of magnitude of p c . More recently, the first and second authors [23] proved that the anisotropic model exhibits a sharp threshold by determining the first term in (1.3).…”

mentioning

confidence: 99%

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Higher order corrections for anisotropic bootstrap percolation

Duminil-Copin

Enter

Hulshof

2017

Probab. Theory Relat. Fields

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We study the critical probability for the metastable phase transition of the two-dimensional anisotropic bootstrap percolation model with (1, 2)-neighbourhood and threshold r = 3. The first order asymptotics for the critical probability were recently determined by the first and second authors. Here we determine the following sharp second and third order asymptotics:We note that the second and third order terms are so large that the first order asymptotics fail to approximate p c even for lattices of size well beyond 10 10 1000 .

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Section: 5mentioning

confidence: 91%

“…The arguments developed in [23] can be applied to prove that the leading order behaviour of p c for the (…”

Section: 5mentioning

confidence: 99%

“…Combining the techniques of [23] with those introduced in this paper, it is possible to prove the following stronger bounds:…”

Section: 5mentioning

confidence: 99%

Section: 7mentioning

confidence: 99%

mentioning

confidence: 99%

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Higher order corrections for anisotropic bootstrap percolation

Duminil-Copin

Enter

Hulshof

2017

Probab. Theory Relat. Fields

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The time of bootstrap percolation with dense initial sets for all thresholds

Bollobás

Smith²,

Uzzell

2014

Random Struct Algorithms

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Abstract. In r-neighbour bootstrap percolation on the vertex set of a graph G, vertices are initially infected independently with some probability p. At each time step, the infected set expands by infecting all uninfected vertices that have at least r infected neighbours. When p is close to 1, we study the distribution of the time at which all vertices become infected. Given t = t(n) = o(log n/ log log n), we prove a sharp threshold result for the probability that percolation occurs by time t in dneighbour bootstrap percolation on the d-dimensional discrete torus T d n . Moreover, we show that for certain ranges of p = p(n), the time at which percolation occurs is concentrated either on a single value or on two consecutive values. We also prove corresponding results for the modified d-neighbour rule.

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Line percolation

Balister¹,

Bollobás

Narayanan³

2018

Random Struct Algorithms

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We study a new geometric bootstrap percolation model, line percolation, on the d-dimensional integer grid [n] d . In line percolation with infection parameter r, infection spreads from a subset A ⊂ [n] d of initially infected lattice points as follows: if there exists an axis-parallel line L with r or more infected lattice points on it, then every lattice point of [n] d on L gets infected, and we repeat this until the infection can no longer spread. The elements of the set A are usually chosen independently, with some density p, and the main question is to determine p c (n, r, d), the density at which percolation (infection of the entire grid) becomes likely. In this paper, we determine p c (n, r, 2) up to a multiplicative factor of 1 + o(1) and p c (n, r, 3) up to a multiplicative constant as n → ∞ for every fixed r ∈ N. We also determine the size of the minimal percolating sets in all dimensions and for all values of the infection parameter. K E Y W O R D Sbootstrap percolation, critical probability, polynomial method INTRODUCTIONBootstrap percolation models and arguments have been used to study a range of phenomena in various areas, ranging from crack formation and the dynamics of glasses to neural nets and economics; see [4,12,20] for a small sample of such applications. In this paper, we introduce and study a new geometric bootstrap percolation model defined on the d-dimensional integer grid [n] d which we call line percolation; here, we write [n] for the set {1, 2, . . . , n}. For v ∈ [n] d , let L(v) denote the set of d Random Struct Alg. 2018;52:597-616.wileyonlinelibrary.com/journal/rsa

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Sharp metastability threshold for an anisotropic bootstrap percolation model

Cited by 33 publications

References 38 publications

Higher order corrections for anisotropic bootstrap percolation

Higher order corrections for anisotropic bootstrap percolation

The time of bootstrap percolation with dense initial sets for all thresholds

Line percolation

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