Physics – Mathematical Physics
Scientific paper
2007-12-13
J. Phys. A: Math. Theor. 41 (2008) 145001
Physics
Mathematical Physics
37 pages, 18 color figures, improved version with several clarifications (mostly in sections 2.1 and 2.4) and one added sectio
Scientific paper
10.1088/1751-8113/41/14/145001
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.
Bouttier Jérémie
Guitter Emmanuel
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