The hull process of the Brownian plane

Abstract

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 $r$. We identify the law of the process $(Z_r)_{r>0}$ as the time-reversal of a continuous-state branching process starting from $+\infty$ at time $-\infty$ and conditioned to hit $0$ at time $0$, and we give an explicit description of the process of hull volumes given the process $(Z_r)_{r>0}$. We obtain an explicit formula for the Laplace transform of the volume of the hull of radius $r$, and we also determine the conditional distribution of this volume given the length of the boundary. Our proofs involve certain new formulas for super-Brownian motion and the Brownian snake in dimension one, which are of independent interest.