## Abstract

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

Original language | English (US) |
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Pages (from-to) | 205-241 |

Number of pages | 37 |

Journal | Annals of Probability |

Volume | 29 |

Issue number | 1 |

DOIs | |

State | Published - Jan 2001 |

## Keywords

- Lattice animals
- Optimization
- Percolation

## ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty