Confluence of geodesic paths and separating loops in large planar quadrangulations

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.

48 pages, 33 color figures. Final version, with one concluding paragraph and one reference added, and several other small corrections

References in corpus (8)

• The topological structure of scaling limits of large planar maps
• Geodesic Distance in Planar Graphs
• Limit of normalized quadrangulations: The Brownian map
• The three-point function of planar quadrangulations
• Blocked edges on Eulerian maps and mobiles: Application to spanning trees, hard particles and the Ising model
• Statistics of geodesics in large quadrangulations
• Combinatorics of bicubic maps with hard particles
• Combinatorics of Hard Particles on Planar Graphs

Cited by in corpus (4)

• Distance statistics in quadrangulations with a boundary, or with a self-avoiding loop
• Bijective enumeration of planar bipartite maps with three tight boundaries, or how to slice pairs of pants
• The three-point function of general planar maps
• Distance statistics in large toroidal maps