Coexistence probability in the last passage percolation model is $6-8\log 2$

Abstract

A competition model on ${\mathbb{N}}^2$ between three clusters and governed by directed last passage percolation is considered. We prove that coexistence, i.e. the three clusters are simultaneously unbounded, occurs with probability $6-8\log 2$. When this happens, we also prove that the central cluster almost surely has a positive density on ${\mathbb{N}}^2$. Our results rely on three couplings, allowing to link the competition interfaces (which represent the borderlines between the clusters) to some particles in the multi-TASEP, and on recent results about collision in the multi-TASEP.

21 pages, 7 figures

References in corpus (1)

• The motion of a second class particle for the tasep starting from a decreasing shock profile

Cited by in corpus (1)

• Multiple geodesics with the same direction