UNIVERSALITY OF THE GEODESIC TREE IN LAST PASSAGE PERCOLATION

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 (N, N), and whose width and length are o(N-2/3) and o(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 delta N-2/3, length up to delta(3/2) N/(log(delta(-1)))(3), and provide lower and upper bounds for the probability that the geodesics agree in that cylinder.