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

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