Volumes in the Uniform Infinite Planar Triangulation: from skeletons to generating functions

TL;DR

This paper introduces a combinatorial method using skeleton decomposition to compute generating functions for volumes within the UIPT, providing explicit formulas and connecting to known scaling limits.

Contribution

It develops a new combinatorial approach to derive explicit generating functions for volumes in the UIPT, extending previous results and linking to scaling limits.

Findings

Explicit formulas for generating functions of hull volumes

Connection between skeleton decomposition and peeling process results

Recovery of known scaling limit results for hull volumes

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 flavor and the main tool is the decomposition of the UIPT into layers, called the skeleton decomposition, introduced by Krikun. In particular, we get explicit formulas for the generating functions of the number of vertices inside hulls (or completed metric balls) centered 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.