Geodesic Distance in Planar Graphs: An Integrable Approach

We discuss the enumeration of planar graphs using bijections with suitably decorated trees, which allow for keeping track of the geodesic distances between faces of the graph. The corresponding generating functions obey non-linear recursion relations on the geodesic distance. These are solved by use of stationary multi-soliton tau-functions of suitable reductions of the KP hierarchy. We obtain a unified formulation of the (multi-) critical continuum limit describing large graphs with marked points at large geodesic distances, and obtain integrable differential equations for the corresponding scaling functions. This provides a continuum formulation of two-dimensional quantum gravity, in terms of the geodesic distance.