### 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) |
---|---|

Pages (from-to) | 205-241 |

Number of pages | 37 |

Journal | Annals of Probability |

Volume | 29 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 2001 |

### Fingerprint

### Keywords

- Lattice animals
- Optimization
- Percolation

### ASJC Scopus subject areas

- Mathematics(all)
- Statistics and Probability

### Cite this

*Annals of Probability*,

*29*(1), 205-241. https://doi.org/10.1214/aop/1008956328

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

Research output: Contribution to journal › Article

*Annals of Probability*, vol. 29, no. 1, pp. 205-241. https://doi.org/10.1214/aop/1008956328

}

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 -