The Annals of Applied Probability

Examples of nonpolygonal limit shapes in i.i.d. first-passage percolation and infinite coexistence in spatial growth models

Michael Damron and Michael Hochman

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Abstract

We construct an edge-weight distribution for i.i.d. first-passage percolation on $\mathbb{Z}^{2}$ whose limit shape is not a polygon and whose extreme points are arbitrarily dense in the boundary. Consequently, the associated Richardson-type growth model can support coexistence of a countably infinite number of distinct species, and the graph of infection has infinitely many ends.

Article information

Source
Ann. Appl. Probab., Volume 23, Number 3 (2013), 1074-1085.

Dates
First available in Project Euclid: 7 March 2013

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1362684854

Digital Object Identifier
doi:10.1214/12-AAP864

Mathematical Reviews number (MathSciNet)
MR3076678

Zentralblatt MATH identifier
1314.60154

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 82B43: Percolation [See also 60K35]

Keywords
First-passage percolation limit shapes extreme points Richardson’s growth model graph of infection

Citation

Damron, Michael; Hochman, Michael. Examples of nonpolygonal limit shapes in i.i.d. first-passage percolation and infinite coexistence in spatial growth models. Ann. Appl. Probab. 23 (2013), no. 3, 1074--1085. doi:10.1214/12-AAP864. https://projecteuclid.org/euclid.aoap/1362684854


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