Statistics of the Voronoi cell perimeter in large bi-pointed maps

We study the statistics of the Voronoi cell perimeter in large bi-pointed planar quadrangulations. Such maps have two marked vertices at a fixed given distance $2s$ and their Voronoi cell perimeter is simply the length of the frontier which separates vertices closer to one marked vertex than to the other. We characterize the statistics of this perimeter as a function of $s$ for maps with a large given volume $N$ both in the scaling limit where $s$ scales as $N^{1/4}$, in which case the Voronoi cell perimeter scales as $N^{1/2}$, and in the local limit where $s$ remains finite, in which case the perimeter scales as $s^2$ for large $s$. The obtained laws are universal and are characteristics of the Brownian map and the Brownian plane respectively.