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Linear Growth for Greedy Lattice Animals

Abstract : Let $d\geq 2$, and let ${X_\bv, \bv\in\ZZ^d$ be an i.i.d.\ family of non-negative random variables with common distribution $F$. Let $N(n)$ be the maximum value of $\sum_\bv\in\xiX_\bv$ over all connected subsets $\xi$ of $\ZZ^d$ of size $n$ which contain the origin. This model of «greedy lattice animals» was introduced by Cox et al.\ (1993) and Gandolfi and Kesten (1994), who showed that if $î X_\bzero^d (\log^+ X_\bzero)^d+\epsilon<\infty$ for some $\epsilon>0$, then $N(n)/n\to N$ a.s.\ and in $\cL_1$ for some $N<\infty$. Using related but partly simpler methods, we derive the same conclusion under the slightly weaker condition that $\int_0^\infty- \big(1-F(x)\big)^1/ddx<\infty$, and show that $N\leq c\int_0^\infty\big(1-F-(x)\big)^1/ddx$ for some constant $c$. We also give analogous results for the related «greedy lattice paths» model.
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Submitted on : Wednesday, May 24, 2006 - 10:22:35 AM
Last modification on : Friday, February 4, 2022 - 3:18:37 AM
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  • HAL Id : inria-00072603, version 1



James B. Martin. Linear Growth for Greedy Lattice Animals. RR-4035, INRIA. 2000. ⟨inria-00072603⟩



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