Multi-ended Markovian triangulations and robust convergence to the UIPT

TL;DR

This paper classifies infinite planar triangulations with a weak Markov property, establishing the uniqueness of the UIPT with average degree 6 and demonstrating the robustness of its convergence under measure perturbations.

Contribution

It provides a complete classification of certain infinite triangulations and proves the robustness of UIPT convergence, extending understanding of random planar maps.

Findings

UIPT is the unique infinite triangulation with average degree 6 satisfying the Markov property

Convergence to UIPT is robust under various measure perturbations

Large deviation estimates for pattern occurrences in uniform triangulations

Abstract

We classify completely the infinite, planar triangulations satisfying a weak spatial Markov property, without assuming one-endedness nor finiteness of vertex degrees. In particular, the Uniform Infinite Planar Triangulation (UIPT) is the only such triangulation with average degree 6. As a consequence, we prove that the convergence of uniform triangulations of the sphere to the UIPT is robust, in the sense that it is preserved under various perturbations of the uniform measure. As another application, we obtain large deviation estimates for the number of occurencies of a pattern in uniform triangulations.