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The structure of unicellular maps, and a connection between maps of positive genus and planar labelled trees
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  • Published: 17 March 2009

The structure of unicellular maps, and a connection between maps of positive genus and planar labelled trees

  • Guillaume Chapuy1 

Probability Theory and Related Fields volume 147, pages 415–447 (2010)Cite this article

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Abstract

A unicellular map is a map which has only one face. We give a bijection between a dominant subset of rooted unicellular maps of given genus and a set of rooted plane trees with distinguished vertices. The bijection applies as well to the case of labelled unicellular maps, which are related to all rooted maps by Marcus and Schaeffer’s bijection. This gives an immediate derivation of the asymptotic number of unicellular maps of given genus, and a simple bijective proof of a formula of Lehman and Walsh on the number of triangulations with one vertex. From the labelled case, we deduce an expression of the asymptotic number of maps of genus g with n edges involving the ISE random measure, and an explicit characterization of the limiting profile and radius of random bipartite quadrangulations of genus g in terms of the ISE.

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Authors and Affiliations

  1. Laboratoire d’Informatique de l’École Polytechnique, 91128, Palaiseau Cedex, France

    Guillaume Chapuy

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  1. Guillaume Chapuy
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Correspondence to Guillaume Chapuy.

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Chapuy, G. The structure of unicellular maps, and a connection between maps of positive genus and planar labelled trees. Probab. Theory Relat. Fields 147, 415–447 (2010). https://doi.org/10.1007/s00440-009-0211-0

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  • Received: 23 June 2008

  • Revised: 20 January 2009

  • Published: 17 March 2009

  • Issue Date: July 2010

  • DOI: https://doi.org/10.1007/s00440-009-0211-0

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Mathematics Subject Classification (2000)

  • 60C05
  • 05A17
  • 05C30
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