Greedy lattice animals: Negative values and unconstrained maxima

Amir Dembo, Alberto Gandolfi, Harry Kesten

Research output: Contribution to journalArticle

Abstract

Let {Xν, ν ∈ ℤd} be i.i.d. random variables, and S(ξ) = ∑ν∈ξ Xν be the weight of a lattice animal ξ. Let Nn = max{S(ξ): \ξ\ = n and ξ contains the origin} and Gn = max{S(ξ) : ξ ⊆ [-n, n]d}. We show that, regardless of the negative tail of the distribution of Xν, if E(X+ν)d(log+(X+ν))d+a < +∞ for some a > 0, then first, limn n-1 Nn = N exists, is finite and constant a.e.; and, second, there is a transition in the asymptotic behavior of Gn depending on the sign of N: if N > 0 then Gn ≈ nd, and if N < 0 then Gn ≤ cn, for some c > 0. The exact behavior of Gn in this last case depends on the positive tail of the distribution of Xν; we show that if it is nontrivial and has exponential moments, then Gn ≈ log n, with a transition from Gn ≈ nd occurring in general not as predicted by large deviations estimates. Finally, if xd (1 - F(x)) → ∞ as x → ∞, then no transition takes place.

Original languageEnglish (US)
Pages (from-to)205-241
Number of pages37
JournalAnnals of Probability
Volume29
Issue number1
DOIs
StatePublished - Jan 1 2001

Fingerprint

Lattice Animals
Tail
I.i.d. Random Variables
Large Deviations
Asymptotic Behavior
Moment
Estimate
Animals

Keywords

  • Lattice animals
  • Optimization
  • Percolation

ASJC Scopus subject areas

  • Mathematics(all)
  • Statistics and Probability

Cite this

Greedy lattice animals : Negative values and unconstrained maxima. / Dembo, Amir; Gandolfi, Alberto; Kesten, Harry.

In: Annals of Probability, Vol. 29, No. 1, 01.01.2001, p. 205-241.

Research output: Contribution to journalArticle

Dembo, Amir ; Gandolfi, Alberto ; Kesten, Harry. / Greedy lattice animals : Negative values and unconstrained maxima. In: Annals of Probability. 2001 ; Vol. 29, No. 1. pp. 205-241.
@article{b0d60b226e8a4eb2babf5664773f8a4d,
title = "Greedy lattice animals: Negative values and unconstrained maxima",
abstract = "Let {Xν, ν ∈ ℤd} be i.i.d. random variables, and S(ξ) = ∑ν∈ξ Xν be the weight of a lattice animal ξ. Let Nn = max{S(ξ): \ξ\ = n and ξ contains the origin} and Gn = max{S(ξ) : ξ ⊆ [-n, n]d}. We show that, regardless of the negative tail of the distribution of Xν, if E(X+ν)d(log+(X+ν))d+a < +∞ for some a > 0, then first, limn n-1 Nn = N exists, is finite and constant a.e.; and, second, there is a transition in the asymptotic behavior of Gn depending on the sign of N: if N > 0 then Gn ≈ nd, and if N < 0 then Gn ≤ cn, for some c > 0. The exact behavior of Gn in this last case depends on the positive tail of the distribution of Xν; we show that if it is nontrivial and has exponential moments, then Gn ≈ log n, with a transition from Gn ≈ nd occurring in general not as predicted by large deviations estimates. Finally, if xd (1 - F(x)) → ∞ as x → ∞, then no transition takes place.",
keywords = "Lattice animals, Optimization, Percolation",
author = "Amir Dembo and Alberto Gandolfi and Harry Kesten",
year = "2001",
month = "1",
day = "1",
doi = "10.1214/aop/1008956328",
language = "English (US)",
volume = "29",
pages = "205--241",
journal = "Annals of Probability",
issn = "0091-1798",
publisher = "Institute of Mathematical Statistics",
number = "1",

}

TY - JOUR

T1 - Greedy lattice animals

T2 - Negative values and unconstrained maxima

AU - Dembo, Amir

AU - Gandolfi, Alberto

AU - Kesten, Harry

PY - 2001/1/1

Y1 - 2001/1/1

N2 - Let {Xν, ν ∈ ℤd} be i.i.d. random variables, and S(ξ) = ∑ν∈ξ Xν be the weight of a lattice animal ξ. Let Nn = max{S(ξ): \ξ\ = n and ξ contains the origin} and Gn = max{S(ξ) : ξ ⊆ [-n, n]d}. We show that, regardless of the negative tail of the distribution of Xν, if E(X+ν)d(log+(X+ν))d+a < +∞ for some a > 0, then first, limn n-1 Nn = N exists, is finite and constant a.e.; and, second, there is a transition in the asymptotic behavior of Gn depending on the sign of N: if N > 0 then Gn ≈ nd, and if N < 0 then Gn ≤ cn, for some c > 0. The exact behavior of Gn in this last case depends on the positive tail of the distribution of Xν; we show that if it is nontrivial and has exponential moments, then Gn ≈ log n, with a transition from Gn ≈ nd occurring in general not as predicted by large deviations estimates. Finally, if xd (1 - F(x)) → ∞ as x → ∞, then no transition takes place.

AB - Let {Xν, ν ∈ ℤd} be i.i.d. random variables, and S(ξ) = ∑ν∈ξ Xν be the weight of a lattice animal ξ. Let Nn = max{S(ξ): \ξ\ = n and ξ contains the origin} and Gn = max{S(ξ) : ξ ⊆ [-n, n]d}. We show that, regardless of the negative tail of the distribution of Xν, if E(X+ν)d(log+(X+ν))d+a < +∞ for some a > 0, then first, limn n-1 Nn = N exists, is finite and constant a.e.; and, second, there is a transition in the asymptotic behavior of Gn depending on the sign of N: if N > 0 then Gn ≈ nd, and if N < 0 then Gn ≤ cn, for some c > 0. The exact behavior of Gn in this last case depends on the positive tail of the distribution of Xν; we show that if it is nontrivial and has exponential moments, then Gn ≈ log n, with a transition from Gn ≈ nd occurring in general not as predicted by large deviations estimates. Finally, if xd (1 - F(x)) → ∞ as x → ∞, then no transition takes place.

KW - Lattice animals

KW - Optimization

KW - Percolation

UR - http://www.scopus.com/inward/record.url?scp=0035533088&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0035533088&partnerID=8YFLogxK

U2 - 10.1214/aop/1008956328

DO - 10.1214/aop/1008956328

M3 - Article

AN - SCOPUS:0035533088

VL - 29

SP - 205

EP - 241

JO - Annals of Probability

JF - Annals of Probability

SN - 0091-1798

IS - 1

ER -