The skeleton of the UIPT, seen from infinity

TL;DR

This paper establishes the strong coalescence of geodesic rays in the UIPT, introduces a new skeleton decomposition from infinity, and relates it to a conditioned Galton-Watson tree, providing new insights and proofs in random triangulation geometry.

Contribution

It introduces a novel skeleton decomposition of the UIPT from infinity and links it to a conditioned Galton-Watson tree, offering a new construction and analysis method.

Findings

Geodesic rays in UIPT coalesce strongly.

Existence of a unique horofunction measuring distances from infinity.

Scaling limits of horohulls are derived, and a new proof of the 2-point function formula is provided.

Abstract

We prove that geodesic rays in the Uniform Infinite Planar Triangulation (UIPT) coalesce in a strong sense using the skeleton decomposition of random triangulations discovered by Krikun. This implies the existence of a unique horofunction measuring distances from infinity in the UIPT. We then use this horofunction to define the skeleton "seen from infinity" of the UIPT and relate it to a simple Galton--Watson tree conditioned to survive, giving a new and particularly simple construction of the UIPT. Scaling limits of perimeters and volumes of horohulls within this new decomposition are also derived, as well as a new proof of the $2$-point function formula for random triangulations in the scaling limit due to Ambj{\o}rn and Watabiki.