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Volumes in the Uniform Infinite Planar Triangulation: From Skeletons to Generating Functions

Part of: Graph theory

Published online by Cambridge University Press:  21 May 2018

LAURENT MÉNARD
LAURENT MÉNARD*
Affiliation:
Modal'X, Université Paris Ouest and LiX, École Polytechnique, 200 avenue de la République, 92000 Nanterre, France (e-mail: laurent.menard@normalesup.org)
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Abstract

We develop a method to compute the generating function of the number of vertices inside certain regions of the Uniform Infinite Planar Triangulation (UIPT). The computations are mostly combinatorial in flavour and the main tool is the decomposition of the UIPT into layers, called the skeleton decomposition, introduced by Krikun [20]. In particular, we get explicit formulas for the generating functions of the number of vertices inside hulls (or completed metric balls) centred around the root, and the number of vertices inside geodesic slices of these hulls. We also recover known results about the scaling limit of the volume of hulls previously obtained by Curien and Le Gall by studying the peeling process of the UIPT in [17].

MSC classification

Secondary: 05C80: Random graphs
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 27 , Issue 6 , November 2018 , pp. 946 - 973
Copyright
Copyright © Cambridge University Press 2018 

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