Confluence of geodesic paths and separating loops in large planar quadrangulations

Details Confluence of geodesic paths and separating loops in large planar quadrangulations Confluence of geodesic paths and separating loops in large planar quadrangulations

2008-11-04

arxiv.org/abs/0811.0509v2

J. Stat. Mech. (2009) P03001

Physics

Mathematical Physics

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

Scientific paper

10.1088/1742-5468/2009/03/P03001

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.

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