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 . 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.
[Che] personal communication
[CKM] The mesoscopic geometry of sparse random maps (to appear in Journal de l’École Polytechnique – Mathématiques)
[CT20b] Ising model on random triangulations of the disk: phase transition (2020) (https://arxiv.org/abs/2003.09343v1)
[Cur19] Peeling random planar maps, Saint-Flour lecture notes (2019)
[Tur20] Interfaces in the vertex-decorated Ising model on random triangulations of the disk (2020) (https://arxiv.org/abs/2003.11012)