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.