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

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    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.
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