Speaker
Jérémie Bouttier
(CEA Saclay)
Description
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.
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.