Uniform infinite planar quadrangulations with a boundary

Nicolas Curien (1) , Grégory Miermont (2)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires
2 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées

We introduce and study the uniform infinite planar quadrangulation (UIPQ) with a boundary via an extension of the construction of [14]. We then relate this object to its simple boundary analog using a pruning procedure. This enables us to study the aperture of these maps, that is, the maximal graph distance between two points on the boundary, which in turn sheds new light on the geometry of the UIPQ. In particular we prove that the self-avoiding walk on the UIPQ is at most diffusive.

Domaines: Probabilités [math.PR]

Nicolas Curien, Grégory Miermont. Uniform infinite planar quadrangulations with a boundary. Random Structures and Algorithms, 2014, 47 (1), pp.30-58. ⟨10.1002/rsa.20531⟩. ⟨hal-01489682⟩