Let d[greater-or-equal, slanted]2, and let be an i.i.d. family of non-negative random variables with common distribution F. Let N(n) be the maximum value of [summation operator]v[set membership, variant][xi]Xv over all connected subsets [xi] of of size n which contain the origin. This model of "greedy lattice animals" was introduced by Cox et al. (Ann. Appl. Probab. 3 (1993) 1151) and Gandolfi and Kesten (Ann. Appl. Probab. 4 (1994) 76), who showed that if for some [var epsilon]>0, then N(n)/n-->N a.s. and in for some N<[infinity]. Using related but partly simpler methods, we derive the same conclusion under the slightly weaker condition that , and show that for some constant c. We also give analogous results for the related "greedy lattice paths" model.
