Geodesic Distance in Planar Graphs

J. Bouttier; P. Di Francesco; E. Guitter

doi:10.1016/s0550-3213(03)00355-9

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arXiv · cond-mat/0303272

Geodesic Distance in Planar Graphs

J. Bouttier·P. Di Francesco·E. Guitter

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Abstract

We derive the exact generating function for planar maps (genus zero fatgraphs) with vertices of arbitrary even valence and with two marked points at a fixed geodesic distance. This is done in a purely combinatorial way based on a bijection with decorated trees, leading to a recursion relation on the geodesic distance. The latter is solved exactly in terms of discrete soliton-like expressions, suggesting an underlying integrable structure. We extract from this solution the fractal dimensions at the various (multi)-critical points, as well as the precise scaling forms of the continuum two-point functions and the probability distributions for the geodesic distance in (multi)-critical random surfaces. The two-point functions are shown to obey differential equations involving the residues of the KdV hierarchy.

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cond-mat.stat-mech hep-lat hep-th math-ph math.CO math.MP nlin.SI

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BibTeX RIS

J. Bouttier, P. Di Francesco, E. Guitter. 2003-03-14. Geodesic Distance in Planar Graphs. https://doi.org/10.1016/s0550-3213(03)00355-9

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