The Annals of Probability

Greedy lattice animals: negative values and unconstrained maxima

Abstract

Let $\{X_v, v \in \mathbb{Z}^d\}$ be i.i.d. random variables, and $S(\xi) = \sum_{v \in \xi} X_v$ be the weight of a lattice animal $\xi$. Let $N_n = \max\{S(\xi) : |\xi| = n$ \text{and $\xi$ contains the origin}\}$and$G_n = \max\{S(\xi) : \xi \subseteq [-n,n]^d\}$. We show that, regardless of the negative tail of the distribution of$X_v$, if$\mathbf{E}( X_v^+)^d (\log^+ X_v^+))^{d+a} < + \infty$for some$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 $G_n \approx n^d$, and if $N < 0$ then $G_n \le cn$, for some $c > 0$. The exact behavior of $G_n$ in this last case depends on the positive tail of the distribution of $X_v$; we show that if it is nontrivial and has exponential moments, then $G_n \approx \log n$, with a transition from $G_n \approx n^d$ occurring in general not as predicted by large deviations estimates. Finally, if $x^d(1 - F(x)) \to \infty$as $x \to \infty$, then no transition takes place.

Article information

Source
Ann. Probab., Volume 29, Number 1 (2001), 205-241.

Dates
First available in Project Euclid: 21 December 2001

Permanent link to this document
https://projecteuclid.org/euclid.aop/1008956328

Digital Object Identifier
doi:10.1214/aop/1008956328

Mathematical Reviews number (MathSciNet)
MR1825148

Zentralblatt MATH identifier
1016.60048

Citation

Dembo, Amir; Gandolfi, Alberto; Kesten, Harry. Greedy lattice animals: negative values and unconstrained maxima. Ann. Probab. 29 (2001), no. 1, 205--241. doi:10.1214/aop/1008956328. https://projecteuclid.org/euclid.aop/1008956328