Geodesic Forests in the Last-Passage Percolation

TL;DR

This paper investigates the structure of geodesic forests in last-passage percolation with exponential weights, revealing how the root location relates to random walk maxima and analyzing the height function's scaling limits.

Contribution

It introduces a novel analysis of point-to-line geodesics, connecting root locations to random walk maxima and characterizing the height function's behavior via variational problems involving the Airy process.

Findings

Root location described by maxima of a random walk

Power law behavior of the height function for flat substrates

Scaling limit characterized by variational problems with the Airy process

Abstract

The aim of this article is to study the forest composed by point-to-line geodesics in the last-passage percolation model with exponential weights. We will show that the location of the root can be described in terms of the maxima of a random walk, whose distribution will depend on the geometry of the substrate (line). For flat substrates, we will get power law behaviour of the height function, study its scaling limit, and describe it in terms of variational problems involving the Airy process.