Volumes in the Uniform Infinite Planar Triangulation: From Skeletons to Generating Functions. (21st May 2018)

• Record Type: Journal Article
• Authors: MÉNARD, LAURENT
• Is Part Of: Combinatorics, probability and computing. Volume 27:Number 6(2018)
• Page Start: 946
• Page End: 973
• Publication Date: 2018-05-21
• Subjects: Primary 60C80, -- Secondary 05C80; Combinatorial analysis -- Periodicals; Probabilities -- Periodicals; Computer science -- Mathematics -- Periodicals; 511.6
• DOI: 10.1017/S0963548318000093
• Languages: English
• ISSNs: 0963-5483

Abstract

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].