Busemann process and semi-infinite geodesics in Brownian last-passage percolation

TL;DR

This paper proves the existence and properties of semi-infinite geodesics in Brownian last-passage percolation, revealing their uniqueness, coalescence, and the absence of bi-infinite geodesics in fixed directions.

Contribution

It constructs a global Busemann process for all points and directions in BLPP, establishing new phenomena in the semi-discrete setting compared to discrete models.

Findings

Existence of semi-infinite geodesics from every point in all directions.

Uniqueness of geodesics for fixed point and direction.

No bi-infinite geodesics in fixed northeast or southwest directions.

Abstract

We prove the existence of semi-infinite geodesics for Brownian last-passage percolation (BLPP). Specifically, on a single event of probability one, there exist semi-infinite geodesics started from every space-time point and traveling in every asymptotic direction. Properties of these geodesics include uniqueness for a fixed initial point and direction, non-uniqueness for fixed direction but random initial points, and coalescence of all geodesics traveling in a common, fixed direction. Along the way, we prove that for fixed northeast and southwest directions, there almost surely exist no bi-infinite geodesics in the given directions. The semi-infinite geodesics are constructed from Busemann functions. Our starting point is a result of Alberts, Rassoul-Agha and Simper that established Busemann functions for fixed points and directions. Out of this, we construct the global process of Busemann functions simultaneously for all initial points and directions, and then the family of semi-infinite Busemann geodesics. The uncountable space of the semi-discrete setting requires extra consideration and leads to new phenomena, compared to discrete models.