The Markovian hyperbolic triangulation

TL;DR

This paper introduces a unique, invariant random tiling of the hyperbolic plane into ideal triangles, exhibiting a natural Markov property and conditional independence, advancing understanding of hyperbolic geometric structures.

Contribution

It constructs and characterizes the only such hyperbolic triangulation with invariance and Markov properties, providing a new model for hyperbolic tilings.

Findings

Unique hyperbolic triangulation constructed

Invariance under Moebius transformations established

Markov property and conditional independence demonstrated

Abstract

We construct and study the unique random tiling of the hyperbolic plane into ideal hyperbolic triangles (with the three corners located on the boundary) that is invariant (in law) with respect to Moebius transformations, and possesses a natural spatial Markov property that can be roughly described as the conditional independence of the two parts of the triangulation on the two sides of the edge of one of its triangles.