Vertex Degrees in Planar Maps

TL;DR

This paper establishes a multi-dimensional central limit theorem for the distribution of vertex degrees in planar maps with restricted degrees, using bijections with mobiles and advanced analytic methods.

Contribution

It introduces a general CLT for vertex degrees in planar maps with arbitrary degree restrictions, extending previous results and employing novel analytic techniques.

Findings

Proves a multi-dimensional CLT for expected vertex degrees

Handles infinite degree sets with refined analytic tools

Discusses potential extensions to higher genus maps

Abstract

We prove a general multi-dimensional central limit theorem for the expected number of vertices of a given degree in the family of planar maps whose vertex degrees are restricted to an arbitrary (finite or infinite) set of positive integers D. Our results rely on a classical bijection with mobiles (objects exhibiting a tree structure), combined with refined analytic tools to deal with the systems of equations on infinite variables that arise. We also discuss some possible extension to maps of higher genus.