2007/01/31 by Pablo A. Ferrari, James B. Martin, Leandro P. R. Pimentel · 2 citations
Decision Sciences · Engineering · Mathematics · Physics and Astronomy · #Game Theory and Applications #Modular Robots and Swarm Intelligence #Theoretical and Computational Physics #math-ph #math.MP #math.PR #msc:60K35 #msc:82B
paper · pdf · doi:10.1214/08-aap542
published as Annals of Applied Probability 2009, Vol. 19, No. 1, 281-317 · Published in at http://dx.doi.org/10.1214/08-AAP542 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)
openalex publication_date 2009/02/01 · arxiv created 2009/03/02 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/01
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.