Geodesic Distance in Planar Graphs
Bouttier, J. ; Di Francesco, P. ; Guitter, E.
arXiv, 0303272 / Harvested from arXiv
Text on arXiv
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.
Publié le : 2003-03-14
Classification:  Condensed Matter - Statistical Mechanics,  High Energy Physics - Lattice,  High Energy Physics - Theory,  Mathematical Physics,  Mathematics - Combinatorics,  Nonlinear Sciences - Exactly Solvable and Integrable Systems 
@article{0303272,
     author = {Bouttier, J. and Di Francesco, P. and Guitter, E.},
     title = {Geodesic Distance in Planar Graphs},
     journal = {arXiv},
     volume = {2003},
     number = {0},
     year = {2003},
     language = {en},
     url = {http://dml.mathdoc.fr/item/0303272}
}

Bouttier, J.; Di Francesco, P.; Guitter, E. Geodesic Distance in Planar Graphs. arXiv, Tome 2003 (2003) no. 0, . http://gdmltest.u-ga.fr/item/0303272/