Linear growth for greedy lattice animals

Let d⩾2, and let {Xv,v∈Zd} be an i.i.d. family of non-negative random variables with common distribution F. Let N(n) be the maximum value of ∑v∈ξXv over all connected subsets ξ of Zd 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 EX0d(log+ X0)d+ε<∞ for some ε>0, then N(n)/n→N a.s. and in L1 for some N<∞. Using related but partly simpler methods, we derive the same conclusion under the slightly weaker condition that ∫0∞(1−F(x))1/d dx<∞, and show that N⩽c∫0∞(1−F(x))1/d dx for some constant c. We also give analogous results for the related “greedy lattice paths” model.