Greedy lattice animals: Negative values and unconstrained maxima
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.
scientific article; zbMATH DE number 1905977
scientific article