Linear Growth for Greedy Lattice Animals
نویسنده
چکیده
Let d 2, and let fXv; v 2 Z d g be an i.i.d. family of non-negative random variables with common distribution F. Let N (n) be the maximum value of P v2 Xv over all connected subsets of Z d of size n which contain the origin. This model of \greedy lattice animals" was introduced by Cox et al. (1993) and Gandoll and Kesten (1994), who showed that if E X d 0 (log + X0) d+ < 1 for some > 0, then N (n)=n ! N a.s. and in L1 for some N < 1. Using related but partly simpler methods, we derive the same conclusion under the slightly weaker condition that R 1 0 ? 1 ? F (x) 1=d dx < 1, and show that N c R 1 0 ? 1 ? F (x) 1=d dx for some constant c. We also give analogous results for the related \greedy lattice paths" model.
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