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Limit laws of planar maps with prescribed vertex degrees

  • G. Collet (a1), M. Drmota (a1) and L. D. Klausner (a1)
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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*Corresponding author. Email: michael.drmota@tuwien.ac.at
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Partially supported by the Austrian Science Fund (FWF) project SFB F50-02 ‘Shape Characteristics of Planar Maps and Planar Graphs’.

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References
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[10] Collet, G. and Fusy, É. (2012) A simple formula for the series of bipartite and quasi-bipartite maps with boundaries. In 24th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2012), DMTCS Proc. AR, pp. 607618.
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Combinatorics, Probability and Computing
  • ISSN: 0963-5483
  • EISSN: 1469-2163
  • URL: /core/journals/combinatorics-probability-and-computing
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