Confluence of geodesic paths and separating loops in large planar quadrangulations

TL;DR

This paper explores the geometry of large random planar quadrangulations, focusing on geodesic paths and separating loops, revealing a universal confluence phenomenon in the Brownian map.

Contribution

It characterizes the universal distribution of confluence lengths of geodesic paths and loops in the Brownian map, highlighting a key geometric feature of random quadrangulations.

Findings

Confluence of geodesic paths occurs with macroscopic length.

Universal probability distribution for confluence lengths.

Distinguishes random quadrangulations from smooth surfaces.

Abstract

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. This is the phenomenon of confluence, which distinguishes the geometry of random quadrangulations from that of smooth surfaces. We characterize the universal probability distribution for the lengths of these common parts.