Planar maps and continued fractions

Planar maps (graphs embedded in the sphere) form a natural model for 
discrete (tessellated) random surfaces, used in the context of 
two-dimensional quantum gravity. Many questions about the geometry of 
random maps can be rephrased as enumeration problems. In this talk, I 
will present an unexpected connection between two such problems.

In the first problem, we consider maps with one boundary, whose 
generating function is the so-called disk amplitude. This quantity is 
well-studied, it is for instance expressible as a matrix integral, and 
computable using Tutte's/loop equations.

In the second problem, we consider maps with two marked points at a 
given distance, whose generating function is the so-called two-point 
function. Though it is one of the simplest metric-related observables, 
much less is known about it.

I will explain that, in a rather general class of maps, the disk 
amplitude and the two-point function are two facets of the same 
quantity, which has to be viewed respectively as a power series and as a 
continuous fraction. I will then explain how the known solution to the 
first problem yields the solution to the second problem.