Annals of Applied Probability

Greedy Lattice Animals II: Linear Growth

Alberto Gandolfi and Harry Kesten

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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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Ann. Appl. Probab., Volume 4, Number 1 (1994), 76-107.

First available in Project Euclid: 19 April 2007

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Zentralblatt MATH identifier


Primary: 60G50: Sums of independent random variables; random walks
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Optimization linear growth subaddivity method of bounded differences


Gandolfi, Alberto; Kesten, Harry. Greedy Lattice Animals II: Linear Growth. Ann. Appl. Probab. 4 (1994), no. 1, 76--107. doi:10.1214/aoap/1177005201.

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See also

  • Part I: J. Theodore Cox, Alberto Gandolfi, Philip S. Griffin, Harry Kesten. Greedy Lattice Animals I: Upper Bounds. Ann. Appl. Probab., Volume 3, Number 4 (1993), 1151--1169.