Abstract
Assign to each site of the integer lattice Z d a real score, sampled according to the same distribution F, independently of the choices made at all other sites. A lattice animal is a finite connected set of sites, with its weight being the sum of the scores at its sites. Let N n be the maximal weight of those lattice animals of size n that contain the origin. Denote by N the almost sure finite constant limit of n -1 N n , which exists under a mild condition on the positive tail of F. We study certain geometrical aspects of the lattice animal with maximal weight among those contained in an n-box where n is large, both in the supercritical phase where N > 0, and in the critical case where N=0.
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