## The Annals of Probability

### Greedy lattice animals: Geometry and criticality

Alan Hammond

#### 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 Nn 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−1Nn, 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.

#### Article information

Source
Ann. Probab., Volume 34, Number 2 (2006), 593-637.

Dates
First available in Project Euclid: 9 May 2006

https://projecteuclid.org/euclid.aop/1147179984

Digital Object Identifier
doi:10.1214/009117905000000693

Mathematical Reviews number (MathSciNet)
MR2223953

Zentralblatt MATH identifier
1097.60081

#### Citation

Hammond, Alan. Greedy lattice animals: Geometry and criticality. Ann. Probab. 34 (2006), no. 2, 593--637. doi:10.1214/009117905000000693. https://projecteuclid.org/euclid.aop/1147179984

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