Volumes in the uniform infinite planar triangulation: from skeletons to generating functions. (English) Zbl 1402.05193
Summary: 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 M. A. Krikun [Zap. Nauchn. Semin. POMI 307, 141–174, 282–283 (2005; Zbl 1074.60027); translation in J. Math. Sci., New York 131, No. 2, 5520–5537 (2005)]. 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 N. Curien and J.-F. Le Gall by studying the peeling process of the UIPT in [Ann. Inst. Henri Poincaré, Probab. Stat. 53, No. 1, 322–357 (2017; Zbl 1358.05255)].
MSC:
| 05C80 | Random graphs (graph-theoretic aspects) |
| 60J80 | Branching processes (Galton-Watson, birth-and-death, etc.) |
| 60F17 | Functional limit theorems; invariance principles |
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