Greedy lattice animals: Negative values and unconstrained maxima
type: Item
instance of: scholarly article
title: Greedy lattice animals: Negative values and unconstrained maxima
zbMATH Open document ID: 1016.60048
author: Amir Dembo
author: Alberto Gandolfi
author: Harry Kesten
published in: The Annals of Probability
publication date: 2003-05-06T00:00:00Z
review text: Let \(\{X_v,\;v\in Z^d\}\) be i.i.d. random variables, and \(S(\xi)=\sum_{v\in\xi}X_v\) be the weight of a lattice animal \(\xi\). Let \(N_n=\max\{S(\xi):|\xi|=n\) and \(\xi\) contains the origin\} and \(G_n=\max\{S(\xi):\xi\subseteq[-n, n]^d\}\). This paper shows that, regardless of the negative tail of the distribution of \(x_v\), if \(E(X_v^+)^d(\log^+(X^+_v))^{d+a}<+\infty\) for some \(a>0\), then first, \(\lim_nn^{-1}N_n=N\) exists, is finite and constant a.e.; and, second, there is a transition in the asymptotic behavior of \(G_n\) depending on the sign of \(N\): if \(N>0\), then \(G_n\approx n^d\), and if \(N<0\), then \(G_n\leq cn\), for some \(c>0\). The exact behavior of \(G_n\) in this last case depends on the positive tail of the distribution of \(X_v\); this paper shows that if it is nontrivial and has exponential moments, then \(G_n\approx \log n\), with a transition from \(G_n\approx n^d\) occurring in general not as predicted by large deviations estimates. Finally, if \(x^d(1-F(x))\rightarrow \infty\) as \(x\rightarrow\infty\), then no transition takes place.
reviewed by: Yuhu Feng
Mathematics Subject Classification ID: 60G50
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 60F10
zbMATH DE Number: 1905977
zbMATH Keywords: optimization
zbMATH Keywords: lattice animals
zbMATH Keywords: percolation
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1008956328
OpenAlex ID: W1593936079
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DOI: 10.1214/AOP/1008956328
description: scientific article; zbMATH DE number 1905977