Greedy lattice animals. I. Upper bounds

Let {X(v): v in Z^d} be an i.i.d. family of positive random variables. For each set 4 of vertices of Z^d, its weight is defined as S(C) = Somma di X(v) per v in C. A greedy lattice animal of size n is a connected subset of Z^d of n vertices, containing the origin, and whose weight is maximal among all such sets. Let N(n) denote this maximal weight. We show that if the expectation of X^d(log(X)^()d+ a is finite for some a > 0, then w.p. 1 N(n) < Mn eventually for some finite constant M. Estimates for the tail of the distribution of N(n ) are also derived.