Inequalities with applications to percolation and reliability

Berg, J. van den; Kesten, Harry

doi:10.1017/s0021900200029326

Cited by 120 publications

(176 citation statements)

References 10 publications

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“…For this special situation (2.44) already follows from van den Berg and Kesten (1985); see also Grimmett (1989, Eq. (2.14)).…”

Section: Proof Of Theoremmentioning

confidence: 75%

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A central limit theorem for “critical” first-passage percolation in two dimensions

Kesten¹,

Zhang²

1997

Probab Theory Relat Fields

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Consider (independent) ÿrst-passage percolation on the edges of Z 2 . Denote the passage time of the edge e in Z 2 by t(e), and assume that P{t(e) = 0} = 1=2, P{0 ¡t(e)¡C 0 } = 0 for some constant C 0 ¿ 0 and that E[t (e)] ¡ ∞ for some ¿4. Denote by b 0; n the passage time from 0 to the halfplane {(x; y): x = n}, and by T (0; nu) the passage time from 0 to the nearest lattice point to nu, for u a unit vector. We prove that there exist constants 0 ¡C 1 , C 2 ¡∞ and n such that C 1 (log n) 1=2 5 n 5 C 2 (log n) 1=2 and such that −1 n [b 0; n − Eb 0; n ] and ( √ 2 n ) −1 [T (0; nu) − ET (0; nu)] converge in distribution to a standard normal variable (as n → ∞, u ÿxed). A similar result holds for the site version of ÿrst-passage percolation on Z 2 , when the common distribution of the passage times {t(v)} of the vertices satisÿes P{t(v) = 0} = 1 − P{t(v) = C 0 } = p c (Z 2 ; site) := critical probability of site percolation on Z 2 , and E[t (u)] ¡ ∞ for some ¿4. Mathematics Subject Classiÿcation (1991): 60K35, 60F05, 82B43

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“…For this special situation (2.44) already follows from van den Berg and Kesten (1985); see also Grimmett (1989, Eq. (2.14)).…”

Section: Proof Of Theoremmentioning

confidence: 75%

“…where G 1 G 2 · · · G r is the event that G 1 ; : : : ; G r occur disjointly, as deÿned in van den Berg and Kesten (1985); see also Grimmett (1989, p. 31). The general inequality (2.44) was only proved recently in Reimer (1994), but we shall apply this with…”

Section: Proof Of Theoremmentioning

confidence: 99%

A central limit theorem for “critical” first-passage percolation in two dimensions

Kesten¹,

Zhang²

1997

Probab Theory Relat Fields

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“…These bounds consist of two-point functions, and are results of applications of the BK inequality [4] and…”

Section: Diagrammatic Bounds On the Expansion Coefficientsmentioning

confidence: 99%

Critical behavior and the limit distribution for long-range oriented percolation. I

Chen

Sakai

2007

Probab. Theory Relat. Fields

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We consider oriented percolation on Zd × Z+ whose bond-occupation probability is pD( · ), where p is the percolation parameter and D is a probability distribution on Zd . Suppose that D(x) decays as |x|−d−α for some α > 0. We prove that the two-point function obeys an infrared bound which implies that various critical exponents take on their respective mean-field values above the upper-critical dimension dc = 2(α ∧2).We also show that, for every k, the Fourier transform of the normalized two-point function at time n,with a proper spatial scaling, has a convergent subsequence to e−c|k|α∧2 for some c > 0

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“…Consider S as a union of |S| 0-dimensional cubes in form (8). Then the sum of the dimensions of the "building blocks" in (8) is 0.…”

Section: Lemma 7 (I) a Subset Of A Hypercube Is Closed Iff It Is A Umentioning

confidence: 99%

“…The lemma is immediate from the van den Berg-Kesten lemma (see [8]) for monotone properties. For a proof of the van den Berg-Kesten conjecture, which is a considerable extension of the van den Berg-Kesten lemma, see Reimer [23].…”

Section: Lemma 10 If S Is Large Enough Thenmentioning

confidence: 99%

Bootstrap percolation on the hypercube

Balogh

Bollobás

2005

Probab. Theory Relat. Fields

103

153

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In the bootstrap percolation on the n-dimensional hypercube, in the initial position each of the 2 n sites is occupied with probability p and empty with probability 1 − p, independently of the state of the other sites. Every occupied site remains occupied for ever, while an empty site becomes occupied if at least two of its neighbours are occupied. If at the end of the process every site is occupied, we say that the (initial) position spans the hypercube. We shall show that there are constants c 1 , c 2 > 0 such that for p(n) ≥ c 1 n 2 2 −2 √ n the probability of spanning tends to 1 as n → ∞, while for p(n) ≤ c 2 n 2 2 −2 √ n the probability tends to 0. Furthermore, we shall show that for each n the transition has a sharp threshold function.

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Inequalities with applications to percolation and reliability

Cited by 120 publications

References 10 publications

A central limit theorem for “critical” first-passage percolation in two dimensions

A central limit theorem for “critical” first-passage percolation in two dimensions

Critical behavior and the limit distribution for long-range oriented percolation. I

Bootstrap percolation on the hypercube

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