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 . 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 . 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.
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Mots clés : Random planar maps, UIPT, spatial Markov property, pattern occurences
@article{AHL_2022__5__1235_0, author = {Budzinski, Thomas}, title = {Multi-ended {Markovian} triangulations and robust convergence to the {UIPT}}, journal = {Annales Henri Lebesgue}, pages = {1235--1259}, publisher = {\'ENS Rennes}, volume = {5}, year = {2022}, doi = {10.5802/ahl.149}, language = {en}, url = {http://www.numdam.org/articles/10.5802/ahl.149/} }
TY - JOUR AU - Budzinski, Thomas TI - Multi-ended Markovian triangulations and robust convergence to the UIPT JO - Annales Henri Lebesgue PY - 2022 SP - 1235 EP - 1259 VL - 5 PB - ÉNS Rennes UR - http://www.numdam.org/articles/10.5802/ahl.149/ DO - 10.5802/ahl.149 LA - en ID - AHL_2022__5__1235_0 ER -
Budzinski, Thomas. Multi-ended Markovian triangulations and robust convergence to the UIPT. Annales Henri Lebesgue, Tome 5 (2022), pp. 1235-1259. doi : 10.5802/ahl.149. http://www.numdam.org/articles/10.5802/ahl.149/
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