Confluence of geodesic paths and separating loops in large planar quadrangulations

Understand

We consider planar quadrangulations with three marked vertices and discuss the geometry of triangles made of three geodesic paths joining them.

• We also study the geometry of minimal separating loops, i.e.
• paths of minimal length among all closed paths passing by one of the three vertices and separating the two others in the quadrangulation.
• We concentrate on the universal scaling limit of large quadrangulations, also known as the Brownian map, where pairs of geodesic paths or minimal separating loops have common parts of non-zero macroscopic length.