Greedy lattice animals: negative values and unconstrained maxima

Abstract

Let $\{X_v, v \in \mathbb{Z}^d\}$ 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 $G_n = \max\{S(\xi) : \xi \subseteq [-n,n]^d\}$. We show that, regardless of the negative tail of the distribution of $X_v$ , if $\mathbf{E}( X_v^+)^d (\log^+ X_v^+))^{d+a} < + \infty$ for some $a>0$, then first, $\lim_n n^{-1} N_n = N$ 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 $G_n \le cn$, for some $c > 0$. The exact behavior of $G_n$ in this last case depends on the positive tail of the distribution of $X_v$; we show 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 $x^d(1 - F(x)) \to \infty \text{ as }x \to \infty$, then no transition takes place.