Competition interfaces and second class particles

Details Competition interfaces and second class particles Competition interfaces and second class particles

2004-06-16

arxiv.org/abs/math/0406333v3

Annals of Probability 2005, Vol. 33, No. 4, 1235-1254

Mathematics

Probability

Published at http://dx.doi.org/10.1214/009117905000000080 in the Annals of Probability (http://www.imstat.org/aop/) by the Ins

Scientific paper

10.1214/009117905000000080

The one-dimensional nearest-neighbor totally asymmetric simple exclusion process can be constructed in the same space as a last-passage percolation model in Z^2. We show that the trajectory of a second class particle in the exclusion process can be linearly mapped into the competition interface between two growing clusters in the last-passage percolation model. Using technology built up for geodesics in percolation, we show that the competition interface converges almost surely to an asymptotic random direction. As a consequence we get a new proof for the strong law of large numbers for the second class particle in the rarefaction fan and describe the distribution of the asymptotic angle of the competition interface.

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