The Busemann Process and Steep Highways in Directed First Passage Percolation

TL;DR

This paper studies the Busemann process in planar directed first passage percolation, establishing its existence, explicit distributions in integrable models, and applications to geodesics, competition interfaces, and particle systems like TASEP.

Contribution

It extends techniques to prove the existence of the Busemann process and derives explicit distributions for integrable models, linking to particle systems and geodesic phenomena.

Findings

Existence of the Busemann process in the studied setting.

Explicit distribution formulas for integrable models.

Applications to semi-infinite geodesics and TASEP invariant measures.

Abstract

We consider the Busemann process in planar directed first passage percolation. We extend existing techniques to establish the existence of the process in our setting and determine its distribution in a number of integrable models. As examples of their utility, we show how these explicit distributions may be used to quantify the semi-infinite geodesics passing through thin rectangles, and the clustering phenomenon observed in competition interface angles. There is a natural connection with various particle systems, and in particular we obtain the multi-class invariant distributions for discrete-time TASEP with parallel updates.