Multi-ended Markovian triangulations and robust convergence to the UIPT

Budzinski, Thomas

Annales Henri Lebesgue, Tome 5 (2022), pp. 1235-1259.

Nous classifions complètement les triangulations planaires infinies vérifiant une propriété de Markov faible, sans supposer que la triangulation n’a qu’un seul bout, ni que les degrés des sommets sont finis. En particulier, l’UIPT (Triangulation Infinie Uniforme du Plan) est la seule de ces triangulations dont le degré moyen vaut $6$. Nous en déduisons que la convergence des triangulations uniformes de la sphère vers l’UIPT est robuste, au sens où elle est préservée par diverses perturbations de la mesure uniforme. Enfin, nous obtenons des estimées de grandes déviations sur le nombre d’occurences d’une petite sous-triangulation dans les triangulations uniformes.

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.

Classification : 60C05
Mots clés : Random planar maps, UIPT, spatial Markov property, pattern occurences

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