The skeleton of the UIPT, seen from infinity
[Le squelette de l’UIPT, vu de l’infini]
Annales Henri Lebesgue, Tome 1 (2018), pp. 87-125.

Nous prouvons une propriété de coalescence forte des rayons géodésiques dans la triangulation infinie uniforme du plan (UIPT) en utilisant la décomposition en squelette de Krikun. Cela implique l’existence d’une unique horofonction mesurant les distances depuis l’infini dans l’UIPT. Cette horodistance permet de définir le squelette vu de l’infini dans l’UIPT qui est intimement relié à un arbre de Galton–Watson critique conditionné à survivre. Comme corollaires, nous donnons une nouvelle construction de l’UIPT, étudions les limites d’échelle du processus des horo-périmètres et horo-volumes, et dérivons une nouvelle preuve de la fonction à deux points en gravité quantique 2D due à Ambjørn et Watabiki.

We prove that geodesic rays in the Uniform Infinite Planar Triangulation (UIPT) coalesce in a strong sense using the skeleton decomposition of random triangulations discovered by Krikun. This implies the existence of a unique horofunction measuring distances from infinity in the UIPT. We then use this horofunction to define the skeleton “seen from infinity” of the UIPT and relate it to a simple Galton–Watson tree conditioned to survive, giving a new and particularly simple construction of the UIPT. Scaling limits of perimeters and volumes of horohulls within this new decomposition are also derived, as well as a new proof of the two point function formula for random triangulations in the scaling limit due to Ambjørn and Watabiki.

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DOI : 10.5802/ahl.3
Classification : 05C80, 60F17
Mots clés : random planar maps, UIPT, skeleton decomposition, geodesic confluence, discrete $3/2$ stable trees
Curien, Nicolas 1 ; Ménard, Laurent 2

1 Laboratoire de Mathématiques d’Orsay, Univ. Paris-Sud, CNRS, Université Paris-Saclay, 91405 Orsay (France)
2 Laboratoire Modal’X, UPL, Univ. Paris Nanterre, 92000 Nanterre (France)
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Curien, Nicolas; Ménard, Laurent. The skeleton of the UIPT, seen from infinity. Annales Henri Lebesgue, Tome 1 (2018), pp. 87-125. doi : 10.5802/ahl.3. http://www.numdam.org/articles/10.5802/ahl.3/

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