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.

arXiv comment34 pages, 14 figures

References in corpus (10)

• Geodesic Distance in Planar Graphs
• Uniform Infinite Planar Triangulations
• Scaling in quantum gravity
• Local limit of labeled trees and expected volume growth in a random quadrangulation
• A view from infinity of the uniform infinite planar quadrangulation
• Planar maps and continued fractions
• Multi-point functions of weighted cubic maps
• Geodesics in Brownian surfaces (Brownian maps)
• Volumes in the Uniform Infinite Planar Triangulation: from skeletons to generating functions
• The distance-dependent two-point function of triangulations: a new derivation from old results

Cited by in corpus (1)

• Characterizing the metric compactification of $L_p$ spaces by random measures