Geodesic distance in planar graphs
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.
- Publication:
-
Nuclear Physics B
- Pub Date:
- July 2003
- DOI:
- arXiv:
- arXiv:cond-mat/0303272
- Bibcode:
- 2003NuPhB.663..535B
- Keywords:
-
- Condensed Matter - Statistical Mechanics;
- High Energy Physics - Lattice;
- High Energy Physics - Theory;
- Mathematical Physics;
- Mathematics - Combinatorics;
- Nonlinear Sciences - Exactly Solvable and Integrable Systems
- E-Print:
- 38 pages, 8 figures, tex, harvmac, epsf
