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November, 1993 Greedy Lattice Animals I: Upper Bounds
J. Theodore Cox, Alberto Gandolfi, Philip S. Griffin, Harry Kesten
Ann. Appl. Probab. 3(4): 1151-1169 (November, 1993). DOI: 10.1214/aoap/1177005277

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.

Citation

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J. Theodore Cox. Alberto Gandolfi. Philip S. Griffin. Harry Kesten. "Greedy Lattice Animals I: Upper Bounds." Ann. Appl. Probab. 3 (4) 1151 - 1169, November, 1993. https://doi.org/10.1214/aoap/1177005277

Information

Published: November, 1993
First available in Project Euclid: 19 April 2007

zbMATH: 0818.60039
MathSciNet: MR1241039
Digital Object Identifier: 10.1214/aoap/1177005277

Subjects:
Primary: 60G50
Secondary: 60K35

Rights: Copyright © 1993 Institute of Mathematical Statistics

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Vol.3 • No. 4 • November, 1993
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