Universality of the geodesic tree in last passage percolation

Abstract

In this paper, we consider the geodesic tree in exponential last passage percolation. We show that for a large class of initial conditions around the origin, the line-to-point geodesic that terminates in a cylinder located around the point $(\mathit{N},\mathit{N})$, and whose width and length are $\mathit{o}({\mathit{N}}^{2/3})$ and $\mathit{o}(\mathit{N})$, respectively, agrees in the cylinder, with the stationary geodesic sharing the same end-point. In the case of the point-to-point model where the geodesic starts from the origin, we consider width $\mathit{\delta }{\mathit{N}}^{2/3}$, length up to ${\mathit{\delta }}^{3/2}\mathit{N}/{(log({\mathit{\delta }}^{-1}))}^3$, and provide lower and upper bounds for the probability that the geodesics agree in that cylinder.