One-point distribution of the geodesic in directed last passage percolation

TL;DR

This paper derives the explicit one-point distribution of the geodesic location and last passage times in directed last passage percolation with exponential weights, including its asymptotic behavior under KPZ scaling.

Contribution

It provides the first explicit distribution formulas for the geodesic location in directed last passage percolation with exponential weights.

Findings

Explicit one-point distribution of the geodesic location and last passage times.

Asymptotic distribution under KPZ scaling.

Limit behavior as parameters tend to infinity.

Abstract

We consider the geodesic of the directed last passage percolation with iid exponential weights. We find the explicit one-point distribution of the geodesic location joint with the last passage times, and its limit as the parameters go to infinity under the KPZ scaling.