Statistics of geodesics in large quadrangulations

Abstract

We study the statistical properties of geodesics, i.e. paths of minimal length, in large random planar quadrangulations. We extend Schaeffer's well-labeled tree bijection to the case of quadrangulations with a marked geodesic, leading to the notion of "spine trees", amenable to a direct enumeration. We obtain the generating functions for quadrangulations with a marked geodesic of fixed length, as well as with a set of "confluent geodesics", i.e. a collection of non-intersecting minimal paths connecting two given points. In the limit of quadrangulations with a large area n, we find in particular an average number 3*2^i of geodesics between two fixed points at distance i>>1 from each other. We show that, for generic endpoints, two confluent geodesics remain close to each other and have an extensive number of contacts. This property fails for a few "exceptional" endpoints which can be linked by truly distinct geodesics. Results are presented both in the case of finite length i and in the scaling limit i ~ n^(1/4). In particular, we give the scaling distribution of the exceptional points.

37 pages, 18 color figures, improved version with several clarifications (mostly in sections 2.1 and 2.4) and one added section (3.1) on ensembles of random quadrangulations

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Cited by in corpus (6)

• The three-point function of planar quadrangulations
• Distance statistics in quadrangulations with a boundary, or with a self-avoiding loop
• Liouville Quantum Gravity and KPZ
• Geodesics in large planar maps and in the Brownian map
• Confluence of geodesic paths and separating loops in large planar quadrangulations
• Asymptotic enumeration of constellations and related families of maps on orientable surfaces