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Limit laws of planar maps with prescribed vertex degrees

Published online by Cambridge University Press:  04 February 2019

G. Collet
Affiliation:
TU Wien, Institute of Discrete Mathematics and Geometry, Wiedner Hauptstraße 8–10, A-1040 Wien, Austria
M. Drmota*
Affiliation:
TU Wien, Institute of Discrete Mathematics and Geometry, Wiedner Hauptstraße 8–10, A-1040 Wien, Austria
L. D. Klausner
Affiliation:
TU Wien, Institute of Discrete Mathematics and Geometry, Wiedner Hauptstraße 8–10, A-1040 Wien, Austria
*
*Corresponding author. Email: michael.drmota@tuwien.ac.at

Abstract

We prove a generalmulti-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.

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Paper
Copyright
© Cambridge University Press 2019 

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Footnotes

Partially supported by the Austrian Science Fund (FWF) project SFB F50-02 ‘Shape Characteristics of Planar Maps and Planar Graphs’.

References

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