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

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. We use this to conclude that for every $\mathbf x,\mathbf y \in \mathbb Z \times \mathbb R$, the Busemann process $\theta \mapsto \mathcal B^{\theta}(\mathbf x,\mathbf y)$ is locally constant except at a discrete set of directions where the process has a jump. The union over all $\mathbf x,\mathbf y$, of this set of directions, denoted $\Theta$, provides detailed information about the uniqueness and coalescence of semi-infinite geodesics. Similar results have been shown for discrete models, but the uncountable set of initial points in BLPP gives rise to new phenomena and requires new methods of proof. One example is that, for every direction $\theta > 0$, there exists a countably infinite set of initial points $\mathbf x$ such that there are two $\theta$-directed geodesics that split at $\mathbf x$ but eventually come back together. As another example of new phenomena, we define the competition interface in BLPP and show that the set of initial points whose competition interface is nontrivial has Hausdorff dimension $\frac{1}{2}$, almost surely. From each of these exceptional initial points, there exists a random direction $\theta \in \Theta$ for which there exists two $\theta$-directed semi-infinite geodesics that split immediately from the initial point and never come back together. Conversely, for every $\theta \in \Theta$ and 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.