## The Annals of Applied Probability

### Greedy Lattice Animals I: Upper Bounds

#### Abstract

Let $\{X_\nu: \nu \in \mathbb{Z}^d\}$ be an i.i.d. family of positive random variables. For each set $\xi$ of vertices of $\mathbb{Z}^d$, its weight is defined as $S(\xi) = \sum_{\nu \in \xi}X_\nu$. A greedy lattice animal of size $n$ is a connected subset of $\mathbb{Z}^d$ of $n$ vertices, containing the origin, and whose weight is maximal among all such sets. Let $N_n$ denote this maximal weight. We show that if the expectation of $X^d_\nu(\log^+ X_\nu)^{d+ a}$ is finite for some $a > 0$, then w.p.1 $N_n \leq Mn$ eventually for some finite constant $M$. Estimates for the tail of the distribution of $N_n$ are also derived.

#### Article information

Source
Ann. Appl. Probab., Volume 3, Number 4 (1993), 1151-1169.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1177005277

Digital Object Identifier
doi:10.1214/aoap/1177005277

Mathematical Reviews number (MathSciNet)
MR1241039

Zentralblatt MATH identifier
0818.60039

JSTOR

#### Citation

Cox, J. Theodore; Gandolfi, Alberto; Griffin, Philip S.; Kesten, Harry. Greedy Lattice Animals I: Upper Bounds. Ann. Appl. Probab. 3 (1993), no. 4, 1151--1169. doi:10.1214/aoap/1177005277. https://projecteuclid.org/euclid.aoap/1177005277