The hull process of the Brownian plane

Understand

We study the random metric space called the Brownian plane, which is closely related to the Brownian map and is conjectured to be the universal scaling limit of many discrete random lattices such as the uniform infinite planar triangulation.

• We obtain a number of explicit distributions for the Brownian plane.
• In particular, we consider, for every $r>0$, the hull of radius $r$, which is obtained by "filling in the holes" in the ball of radius $r$ centered at the root.
• We introduce a quantity $Z_r$ which is interpreted as the (generalized) length of the boundary of the hull of radius $r$.