In Brownian last-passage percolation (BLPP), the Busemann functions
are indexed by
two points
, and a
direction parameter
.
We derive the joint distribution of Busemann functions across all directions.
The set of directions where the Busemann process is discontinuous, denoted
by ,
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
,
there exists a countably infinite set of initial points
such that there
exist two
-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
.
From each of these exceptional points, there exists a random direction
for which there
exist two
-directed
semi-infinite geodesics that split immediately and never meet again. Conversely, when
, from every initial
point
, there exist
two
-directed
semi-infinite geodesics that eventually separate. Whenever
, all
-directed
semi-infinite geodesics coalesce.