Geodesic Distance in Planar Graphs and Solitons

by Di Francesco, Philippe

Abstract

We solve the problem of enumeration of planar graphs with two marked points at a fixed geodesic distance, by use of an equivalence between planar graphs and decorated trees. The result takes the form of a simple recursion relation on the geodesic distance. The explicit solutions involve soliton-like solutions of discrete KP equations. We obtain fractal dimensions and scaling functions for all multicritical cases of 2D quantum gravity.