Greedy lattice animals III: negative values and unconstrained maxima.

Let {X-upsilon, upsilon epsilon Z(d)) be i.i.d. random variables, and S(xi) = Sigma (upsilon epsilon xi) X-upsilon 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 subset of or equal to [-n, n](d)). We show that, regardless of the negative tail of the distribution of X-upsilon, if E(X-upsilon(+))(d) (log(+)(X-upsilon(+))(d+a) < +infinity or 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) approximate to n(d), and if N < 0 then G(n) less than or equal to 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-upsilon; we show that if it is nontrivial and has exponential moments, then G(n) approximate to log n, with a transition from G(n) approximate to n(d) occurring in general not as predicted by large deviations estimates. Finally, if x(d)(l - F(x)) --> infinity as x --> infinity, then no transition takes place.