Geodesic Distance in Planar Graphs

TL;DR

This paper derives an exact combinatorial generating function for planar maps with two marked points at a fixed geodesic distance, revealing integrable structures and scaling behaviors in random surfaces.

Contribution

It introduces a combinatorial approach to compute geodesic distances in planar graphs, connecting to integrable systems and providing explicit scaling forms.

Findings

Exact generating functions for planar maps with fixed geodesic distance

Identification of integrable structures related to the KdV hierarchy

Explicit scaling forms and fractal dimensions at critical points

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.