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February, 1994 Greedy Lattice Animals II: Linear Growth
Alberto Gandolfi, Harry Kesten
Ann. Appl. Probab. 4(1): 76-107 (February, 1994). DOI: 10.1214/aoap/1177005201

Abstract

Let $\{X_\nu: \nu \in \mathbb{Z}^d\}$ be i.i.d. positive random variables and define $M_n = \max\big\{\sum_{\nu \in \pi}X_\nu: \pi \text{a self-avoiding path of length} n \text{starting at the origin}\big\}$, $N_n = \max\big\{\sum_{\nu \in \xi}X_\nu:\xi \text{a lattice animal of size} n \text{containing the origin}\big\}$. In a preceding paper it was shown that if $E\{X^d_0(\log^+ X_0)^{d+a}\} < \infty$ for some $a > 0$, then there exists some constant $C$ such that w.p.1, $0 \leq M_n \leq N_n \leq Cn$ for all large $n$. In this part we improve this result by showing that, in fact, there exist constants $M,N < \infty$ such that w.p.1, $M_n/n \rightarrow M$ and $N_n/n \rightarrow N$.

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Alberto Gandolfi. Harry Kesten. "Greedy Lattice Animals II: Linear Growth." Ann. Appl. Probab. 4 (1) 76 - 107, February, 1994. https://doi.org/10.1214/aoap/1177005201

Information

Published: February, 1994
First available in Project Euclid: 19 April 2007

zbMATH: 0824.60100
MathSciNet: MR1258174
Digital Object Identifier: 10.1214/aoap/1177005201

Subjects:
Primary: 60G50
Secondary: 60K35

Keywords: linear growth , method of bounded differences , optimization , subaddivity

Rights: Copyright © 1994 Institute of Mathematical Statistics

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Vol.4 • No. 1 • February, 1994
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