Limit Laws of Planar Maps with Prescribed Vertex Degrees

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 framework for analyzing vertex degree distributions in planar maps with arbitrary degree restrictions, extending previous results to infinite degree sets.

Findings

Proves a multi-dimensional CLT for vertex degrees in planar maps.

Develops analytic techniques for systems with infinitely many variables.

Discusses extensions to higher genus and weighted 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 possible extensions to maps of higher genus and to weighted maps.