Sharp metastability threshold for an anisotropic bootstrap percolation model

H. Duminil-Copin; A.C.D.  van Enter

doi:10.1214/11-AOP722

Sharp metastability threshold for an anisotropic bootstrap percolation model

H. Duminil-Copin^*, A.C.D. van Enter

^*Corresponding author for this work

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24 Citations (Scopus)

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Abstract

Bootstrap percolation models have been extensively studied during the two past decades. In this article, we study the following "anisotropic" boot-strap percolation model: the neighborhood of a point (m, n) is the set

{(m + 2, n), (m + 1, n), (m, n + 1), (m - 1, n), (m - 2, n), (m, n - 1)}.

At time 0, sites are occupied with probability p. At each time step, sites that are occupied remain occupied, while sites that are not occupied become occupied if and only if three of more sites in their neighborhood are occupied. We prove that it exhibits a sharp metastability threshold. This is the first mathematical proof of a sharp threshold for an anisotropic bootstrap percolation model.

Original language	English
Pages (from-to)	1218-1242
Number of pages	25
Journal	Annals of probability
Volume	41
Issue number	3A
DOIs	https://doi.org/10.1214/11-AOP722
Publication status	Published - May-2013

Keywords

Bootstrap percolation
sharp threshold
anisotropy
metastability
CELLULAR-AUTOMATA
3 DIMENSIONS
BEHAVIOR
RULE

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24 Citations
1 Erratum

Erratum to: Sharp metastability threshold for an anisotropic bootstrap percolation model (vol 41, pg 1218, 2013)
Duminil-Copin, H. & van Enter, A., Mar-2016, In: Annals of probability. 44, 2, p. 1599-1599 1 p.
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title = "Sharp metastability threshold for an anisotropic bootstrap percolation model",

abstract = "Bootstrap percolation models have been extensively studied during the two past decades. In this article, we study the following {"}anisotropic{"} boot-strap percolation model: the neighborhood of a point (m, n) is the set{(m + 2, n), (m + 1, n), (m, n + 1), (m - 1, n), (m - 2, n), (m, n - 1)}.At time 0, sites are occupied with probability p. At each time step, sites that are occupied remain occupied, while sites that are not occupied become occupied if and only if three of more sites in their neighborhood are occupied. We prove that it exhibits a sharp metastability threshold. This is the first mathematical proof of a sharp threshold for an anisotropic bootstrap percolation model.",

keywords = "Bootstrap percolation, sharp threshold, anisotropy, metastability, CELLULAR-AUTOMATA, 3 DIMENSIONS, BEHAVIOR, RULE",

author = "H. Duminil-Copin and {van Enter}, A.C.D.",

note = "Relation: http://www.rug.nl/research/jbi/ Rights: University of Groningen, Johann Bernoulli Institute for Mathematics and Computer Science",

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pages = "1218--1242",

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TY - JOUR

T1 - Sharp metastability threshold for an anisotropic bootstrap percolation model

AU - Duminil-Copin, H.

AU - van Enter, A.C.D.

N1 - Relation: http://www.rug.nl/research/jbi/ Rights: University of Groningen, Johann Bernoulli Institute for Mathematics and Computer Science

PY - 2013/5

Y1 - 2013/5

N2 - Bootstrap percolation models have been extensively studied during the two past decades. In this article, we study the following "anisotropic" boot-strap percolation model: the neighborhood of a point (m, n) is the set{(m + 2, n), (m + 1, n), (m, n + 1), (m - 1, n), (m - 2, n), (m, n - 1)}.At time 0, sites are occupied with probability p. At each time step, sites that are occupied remain occupied, while sites that are not occupied become occupied if and only if three of more sites in their neighborhood are occupied. We prove that it exhibits a sharp metastability threshold. This is the first mathematical proof of a sharp threshold for an anisotropic bootstrap percolation model.

AB - Bootstrap percolation models have been extensively studied during the two past decades. In this article, we study the following "anisotropic" boot-strap percolation model: the neighborhood of a point (m, n) is the set{(m + 2, n), (m + 1, n), (m, n + 1), (m - 1, n), (m - 2, n), (m, n - 1)}.At time 0, sites are occupied with probability p. At each time step, sites that are occupied remain occupied, while sites that are not occupied become occupied if and only if three of more sites in their neighborhood are occupied. We prove that it exhibits a sharp metastability threshold. This is the first mathematical proof of a sharp threshold for an anisotropic bootstrap percolation model.

KW - Bootstrap percolation

KW - sharp threshold

KW - anisotropy

KW - metastability

KW - CELLULAR-AUTOMATA

KW - 3 DIMENSIONS

KW - BEHAVIOR

KW - RULE

U2 - 10.1214/11-AOP722

DO - 10.1214/11-AOP722

M3 - Article

SN - 0091-1798

VL - 41

SP - 1218

EP - 1242

JO - Annals of probability

JF - Annals of probability

IS - 3A

ER -

Sharp metastability threshold for an anisotropic bootstrap percolation model

Abstract

Keywords

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Research output

Erratum to: Sharp metastability threshold for an anisotropic bootstrap percolation model (vol 41, pg 1218, 2013)

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