The Annals of Applied Probability

Greedy Lattice Animals I: Upper Bounds

J. Theodore Cox, Alberto Gandolfi, Philip S. Griffin, and Harry Kesten

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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

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

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 lattice animals self-avoiding paths spanning trees


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.

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See also

  • Part II: Alberto Gandolfi, Harry Kesten. Greedy Lattice Animals II: Linear Growth. Ann. Appl. Probab., Volume 4, Number 1 (1994), 76--107.