A view from infinity of the uniform infinite planar quadrangulation

Nicolas Curien (1) , Laurent Ménard (2) , Grégory Miermont (3)

1 IUF - Institut universitaire de France
2 MODAL'X - Modélisation aléatoire de Paris X
3 DMA - Département de Mathématiques et Applications - ENS Paris

We introduce a new construction of the Uniform Infinite Planar Quadrangulation (UIPQ). Our approach is based on an extension of the Cori-Vauquelin-Schaeffer mapping in the context of infinite trees, in the spirit of previous work. However, we release the positivity constraint on the labels of trees which was imposed in these references, so that our construction is technically much simpler. This approach allows us to prove the conjectures of Krikun pertaining to the "geometry at infinity" of the UIPQ, and to derive new results about the UIPQ, among which a fine study of infinite geodesics.

Domaines: Probabilités [math.PR]

Nicolas Curien, Laurent Ménard, Grégory Miermont. A view from infinity of the uniform infinite planar quadrangulation. ALEA : Latin American Journal of Probability and Mathematical Statistics, 2013, 10 (1), pp.45-88. ⟨hal-01700905⟩