Greedy lattice animals: geometry and criticality (with an Appendix)
Abstract
Assign to each site of the integer lattice $\Zd$ 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
.