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 : 2003-07-28
Classification:
PACS: 02.10.Eb; 05.50.+q; 04.60.Nc; 64.60.Kw,
[MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO],
[NLIN.NLIN-SI]Nonlinear Sciences [physics]/Exactly Solvable and Integrable Systems [nlin.SI],
[PHYS.COND.CM-SM]Physics [physics]/Condensed Matter [cond-mat]/Statistical Mechanics [cond-mat.stat-mech],
[PHYS.HLAT]Physics [physics]/High Energy Physics - Lattice [hep-lat],
[PHYS.HTHE]Physics [physics]/High Energy Physics - Theory [hep-th],
[PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph],
[MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph]
@article{hal-00586662,
author = {Bouttier, J\'er\'emie and Di Francesco, Philippe and Guitter, Emmanuel},
title = {Geodesic distance in planar graphs},
journal = {HAL},
volume = {2003},
number = {0},
year = {2003},
language = {en},
url = {http://dml.mathdoc.fr/item/hal-00586662}
}
Bouttier, Jérémie; Di Francesco, Philippe; Guitter, Emmanuel. Geodesic distance in planar graphs. HAL, Tome 2003 (2003) no. 0, . http://gdmltest.u-ga.fr/item/hal-00586662/