Confluence of geodesic paths and separating loops in large planar quadrangulations

Abstract

We consider planar quadrangulations with three marked vertices and discuss the geometryof triangles made of three geodesic paths joining them. We also study the geometry ofminimal separating loops, i.e. paths of minimal length among all closed paths passing byone of the three vertices and separating the two others in the quadrangulation. Weconcentrate on the universal scaling limit of large quadrangulations, also known as theBrownian map, where pairs of geodesic paths or minimal separating loops have commonparts of non-zero macroscopic length. This is the phenomenon of confluence, whichdistinguishes the geometry of random quadrangulations from that of smooth surfaces. Wecharacterize the universal probability distribution for the lengths of these commonparts.