The skeleton of the UIPT, seen from infinity

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 Ambjørn and Watabiki.

34 pages, 14 figures