2003/03/14 by J. Bouttier, P. Di Francesco, E. Guitter · 1 citation
Physics and Astronomy · Mathematics · #cond-mat.stat-mech #hep-lat #hep-th #math-ph #math.CO #math.MP #nlin.SI
paper · pdf · doi:10.1016/s0550-3213(03)00355-9
published as Nucl.Phys. B663 (2003) 535-567 · 38 pages, 8 figures, tex, harvmac, epsf
arxiv created 2003/03/14 · arxiv updated 2010/04/05
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.