Metastability Thresholds for Anisotropic Bootstrap Percolation in Three Dimensions

Enter, Aernout van; Fey, Anne

doi:10.1007/s10955-012-0455-4

Cited by 15 publications

(24 citation statements)

References 29 publications

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“…A number of variations of the bootstrap process described above have been considered. Holroyd [21,22] [8,13,14,15,26]. Similar but weaker results about the threshold behaviour of a very general class of update rules on Z 2 were proved in [3,7,9].…”

Section: Introductionmentioning

confidence: 64%

An Improved Upper Bound for Bootstrap Percolation in All Dimensions

Uzzell

2019

Combinator. Probab. Comp.

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In r-neighbor bootstrap percolation on the vertex set of a graph G, a set A of initially infected vertices spreads by infecting, at each time step, all uninfected vertices with at least r previously infected neighbors. When the elements of A are chosen independently with some probability p, it is natural to study the critical probability pc(G, r) at which it becomes likely that all of V (G) will eventually become infected. Improving a result of Balogh, Bollobás, and Morris, we give a bound on the second term in the expansion of the critical probability when G = [n] d and d ≥ r ≥ 2. We show that for all d ≥ r ≥ 2 there exists a constant c d,r > 0 such that if n is sufficiently large, thenwhere λ(d, r) is an exact constant and log (k) (n) denotes the k-times iterated natural logarithm of n.1 It follows that there exists α > 0 such that the right-hand side of (A.6) is at least α(log n) 1/2 − d log n, which is what we wanted.

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Section: Introductionmentioning

confidence: 64%

An Improved Upper Bound for Bootstrap Percolation in All Dimensions

Uzzell

2019

Combinator. Probab. Comp.

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“…The threshold was also determined in the general case r = a 1 + a 2 by van Enter and Fey [13] and the proof can be extended to all a 2 + 1 r a 1 + a 2 : as p → 0,…”

Section: Introductionmentioning

confidence: 85%

“…The base case is (a) and we will focus on that again, whose a straightforward application of Aizenman-Lebowitz Lemma (for the isotropic case, see for instance, (3.30) in [8]). We will prove Proposition 3.12(a) in the anisotropic case, and the proof is similar to that of Lemma 5.4 in [13] (with i = a only), which does not seem to be complete.…”

Section: Lower Boundsmentioning

confidence: 93%

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The $d$-dimensional bootstrap percolation models with threshold at least double exponential

Daniel¹

2022

Preprint

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Consider a p-random subset A of initially infected vertices in the discrete cube [L] d , and assume that the neighbourhood of each vertex consists of the a i nearest neighbours in the ±e i -directions for each i ∈ {1, 2, . . . , d}, where a 1 a 2 . . . a d . Suppose we infect any healthy vertex v ∈ [L] d already having r infected neighbours, and that infected sites remain infected forever. In this paper we determine the (d − 1)-times iterated logarithm of the critical length for percolation up to a constant factor, for all d-tuples (a 1 , . . . , a d ) and all r ∈ {a 2Moreover, we reduce the problem of determining this (coarse) threshold for all d 3 and all r ∈ {a d + 1, . . . , a 1 + a 2 + • • • + a d }, to that of determining the threshold for all d 3 and all r ∈ {a d + 1, . . . , a d−1 + a d }. L c (N a 1 ,...,a d r , p) := min{L ∈ N : P p ([A] = [L] d ) 1/2}.

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“…In higher dimensions it turns out that the leading behaviour is ruled by the two "easiest" growth directions [29].…”

Section: Bootstrap Percolation Modelsmentioning

confidence: 99%

Scaling and Inverse Scaling in Anisotropic Bootstrap Percolation

Enter

2018

Emergence, Complexity and Computation

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In bootstrap percolation it is known that the critical percolation threshold tends to converge slowly to zero with increasing system size, or, inversely, the critical size diverges fast when the percolation probability goes to zero. To obtain higher-order terms (that is, sharp and sharper thresholds) for the percolation threshold in general is a hard question. In the case of two-dimensional anisotropic models, sometimes such correction terms can be obtained from inversion in a relatively simple manner.

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Metastability Thresholds for Anisotropic Bootstrap Percolation in Three Dimensions

Cited by 15 publications

References 29 publications

An Improved Upper Bound for Bootstrap Percolation in All Dimensions

An Improved Upper Bound for Bootstrap Percolation in All Dimensions

The $d$-dimensional bootstrap percolation models with threshold at least double exponential

Scaling and Inverse Scaling in Anisotropic Bootstrap Percolation

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