Multiple geodesics with the same direction

TL;DR

This paper investigates the asymptotic directions of infinite geodesics in a directed last-passage percolation model, proving that no more than two geodesics share the same direction with probability one.

Contribution

It completes the study of geodesic directions by showing that multiple geodesics with the same direction are limited to at most two, using recent results and local modifications.

Findings

No more than two geodesics share the same direction with probability one.

The study extends previous work on geodesic asymptotics in LPP models.

Utilizes recent theoretical results and local modification techniques.

Abstract

The directed last-passage percolation (LPP) model with independent exponential times is considered. We complete the study of asymptotic directions of infinite geodesics, started by Ferrari and Pimentel \cite{FP}. In particular, using a recent result of \cite{CH2} and a local modification argument, we prove there is no (random) direction with more than two geodesics with probability 1.