Universality of the geodesic tree in last passage percolation

TL;DR

This paper demonstrates the universality of the geodesic tree in exponential last passage percolation, showing that geodesics with different initial conditions tend to agree in large regions, revealing a universal behavior.

Contribution

It establishes the universality of geodesic trees in exponential last passage percolation for a broad class of initial conditions, including precise bounds for agreement regions.

Findings

Geodesics with different initial conditions agree in cylinders of specified sizes.

The agreement probability bounds are provided for point-to-point geodesics.

Universal behavior of geodesic trees is confirmed in exponential 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.