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

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.