Greedy lattice animals: Negative values and unconstrained maxima

Let {X_ν, ν ∈ ℤ^d} be i.i.d. random variables, and S(ξ) = ∑_ν∈ξ X_ν be the weight of a lattice animal ξ. Let N_n = max{S(ξ): \ξ\ = n and ξ contains the origin} and G_n = max{S(ξ) : ξ ⊆ [-n, n]^d}. We show that, regardless of the negative tail of the distribution of X_ν, if E(X⁺_ν)^d(log⁺(X⁺ _ν))^d+a < +∞ 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 ≈ n^d, and if N < 0 then G_n ≤ 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_ν; we show that if it is nontrivial and has exponential moments, then G_n ≈ log n, with a transition from G_n ≈ n^d occurring in general not as predicted by large deviations estimates. Finally, if x^d (1 - F(x)) → ∞ as x → ∞, then no transition takes place.