Linear growth for greedy lattice animals

Linear Growth for Greedy Lattice Animals

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 ...

Greedy Lattice Animals: Geometry and Criticality

Assign to each site of the integer lattice Z 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 Nn be the maximal weight of those lattice animals of size n that contain the origin. Denote by N the almost sure finite co...

Greedy Lattice Animals : Geometry and Criticality ( with an Appendix ) Alan

Assign to each site of the integer lattice Z 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 Nn be the maximal weight of those lattice animals of size n that contain the origin. Denote by N the almost sure finite co...

Greedy Lattice Animals: Negative Values and Unconstrained Maxima

Expansion in High Dimension for the Growth Constants of Lattice Trees and Lattice Animals

We compute the first three terms of the 1/d expansions for the growth constants and onepoint functions of nearest-neighbour lattice trees and lattice (bond) animals on the integer lattice Zd, with rigorous error estimates. The proof uses the lace expansion, together with a new expansion for the one-point functions based on inclusion-exclusion. AMS 2010 Subject Classification: 60K35, 82B41 1 Mai...