The Annals of Applied Probability

A phase transition for competition interfaces

Pablo A. Ferrari, James B. Martin, and Leandro P. R. Pimentel

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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.

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Ann. Appl. Probab., Volume 19, Number 1 (2009), 281-317.

First available in Project Euclid: 20 February 2009

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Zentralblatt MATH identifier

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

Asymmetric simple exclusion second-class particle Burgers equation rarefaction fan last-passage percolation competition interface


Ferrari, Pablo A.; Martin, James B.; Pimentel, Leandro P. R. A phase transition for competition interfaces. Ann. Appl. Probab. 19 (2009), no. 1, 281--317. doi:10.1214/08-AAP542.

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