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 of width $o(N^{2/3})$ and length $o(N)$ agrees in the cylinder, with the stationary geodesic sharing the same end point. In the case of the point-to-point model, we consider width $\delta N^{2/3}$ and length up to ${\delta }^{3/2}N/(\log ({\delta }^{-1}){)}^3$ and provide lower and upper bound for the probability that the geodesics agree in that cylinder.

39 pages