Multi-ended Markovian triangulations and robust convergence to the UIPT

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.

The mesoscopic geometry of sparse random maps

“…-In the planar case, by [Ste18] uniform trivalent plane maps converge in distribution in the local topology to the dual map of the Uniform Infinite Planar Triangulation (UIPT) of type 1. This result has recently been extended to the case of essentially trivalent plane maps (i.e., when the defect number is negligible compared to the size of the map) by Budzinski [Bud21].…”

“…Acknowledgements. -We are grateful to Thomas Budzinski for sharing early stages of his work [Bud21], Éric Fusy for the reference [BCR93], as well as Charles Bordenave and Bram Petri for the pointer to [Wor99, Th. 2.19].…”

“…The first step towards a proof of Conjecture 19 would be to prove the convergence of random trivalent maps with a small defect number compared to the number of edges to the Brownian sphere, once rescaled by the fourth root of the number of edges. This would complement the local point of view of [Bud21]. Let us mention that, because we were especially interested in the mesoscopic behaviour of the maps, we assumed throughout this work that f n = o(n), whereas in [KM21b], convergence of uniformly chosen bipartite maps to the Brownian sphere is shown whenever f n , n−f n → ∞.…”