Greedy lattice animals: geometry and criticality (with an Appendix)
Abstract
Description
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 $N = 0$.
46 pages. Contains submitted paper, as well as an appendix. In the appendix, greedy lattice animals of constrained size are studied, and an alternative proof of Theorem 1.3 is given
46 pages. Contains submitted paper, as well as an appendix. In the appendix, greedy lattice animals of constrained size are studied, and an alternative proof of Theorem 1.3 is given