Global structure of semi-infinite geodesics and competition interfaces in Brownian last-passage percolation

TL;DR

This paper analyzes the structure of semi-infinite geodesics and competition interfaces in Brownian last-passage percolation, revealing new phenomena related to geodesic uniqueness, coalescence, and the Hausdorff dimension of nontrivial interfaces.

Contribution

It derives the joint distribution of Busemann functions across all directions and characterizes the set of directions with discontinuous Busemann processes, unveiling novel behaviors in BLPP.

Findings

Existence of countably infinite geodesics splitting and coalescing in each direction.

The set of initial points with nontrivial competition interfaces has Hausdorff dimension 1/2.

Presence of directions where geodesics split immediately and never meet again.

Abstract

In Brownian last-passage percolation (BLPP), the Busemann functions $\mathcal B^{\theta}(\mathbf x,\mathbf y)$ are indexed by two points $\mathbf x,\mathbf y \in \mathbb Z \times \mathbb R$, and a direction parameter $\theta > 0$. We derive the joint distribution of Busemann functions across all directions. The set of directions where the Busemann process is discontinuous, denoted $\Theta$, provides detailed information about the uniqueness and coalescence of semi-infinite geodesics. The uncountable set of initial points in BLPP gives rise to new phenomena not seen in discrete models. For example, in every direction $\theta > 0$, there exists a countably infinite set of initial points $\mathbf x$ such that there exist two $\theta$-directed geodesics that split but eventually coalesce. Further, we define the competition interface in BLPP and show that the set of initial points whose competition interface is nontrivial has Hausdorff dimension $\frac{1}{2}$. From each of these exceptional points, there exists a random direction $\theta \in \Theta$ for which there exists two $\theta$-directed semi-infinite geodesics that split immediately and never meet again. Conversely, when $\theta \in \Theta$, from every initial point $\mathbf x \in \mathbb Z \times \mathbb R$, there exists two $\theta$-directed semi-infinite geodesics that eventually separate. Whenever $\theta \notin \Theta$, all $\theta$-directed semi-infinite geodesics coalesce.