Greedy lattice animals: Geometry and criticality

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⁻¹N_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.