Open Access
February 2009 A phase transition for competition interfaces
Pablo A. Ferrari, James B. Martin, Leandro P. R. Pimentel
Ann. Appl. Probab. 19(1): 281-317 (February 2009). DOI: 10.1214/08-AAP542

Abstract

We study the competition interface between two growing clusters in a growth model associated to last-passage percolation. When the initial unoccupied set is approximately a cone, we show that this interface has an asymptotic direction with probability 1. The behavior of this direction depends on the angle θ of the cone: for θ≥180°, the direction is deterministic, while for θ<180°, it is random, and its distribution can be given explicitly in certain cases. We also obtain partial results on the fluctuations of the interface around its asymptotic direction. The evolution of the competition interface in the growth model can be mapped onto the path of a second-class particle in the totally asymmetric simple exclusion process; from the existence of the limiting direction for the interface, we obtain a new and rather natural proof of the strong law of large numbers (with perhaps a random limit) for the position of the second-class particle at large times.

Citation

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Pablo A. Ferrari. James B. Martin. Leandro P. R. Pimentel. "A phase transition for competition interfaces." Ann. Appl. Probab. 19 (1) 281 - 317, February 2009. https://doi.org/10.1214/08-AAP542

Information

Published: February 2009
First available in Project Euclid: 20 February 2009

zbMATH: 1185.60109
MathSciNet: MR2498679
Digital Object Identifier: 10.1214/08-AAP542

Subjects:
Primary: 60K35
Secondary: 82B

Keywords: Asymmetric simple exclusion , Burgers equation , competition interface , Last-passage percolation , Rarefaction fan , Second-class particle

Rights: Copyright © 2009 Institute of Mathematical Statistics

Vol.19 • No. 1 • February 2009
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