Limits of the boundary of random planar maps

Loïc Richier (1)

1 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées

We discuss asymptotics for the boundary of critical Boltzmann planar maps under the assumption that the distribution of the degree of a typical face is in the domain of attraction of a stable distribution with parameter $\alpha \in (1,2)$. First, in the dense phase corresponding to $\alpha\in(1,3/2)$, we prove that the scaling limit of the boundary is the random stable looptree with parameter $1/(\alpha-1/2)$. Second, we show the existence of a phase transition through local limits of the boundary: in the dense phase, the boundary is tree-like, while in the dilute phase corresponding to $\alpha\in(3/2,2)$, it has a component homeomorphic to the half-plane. As an application, we identify the limits of loops conditioned to be large in the rigid $O(n)$ loop model on quadrangulations, proving thereby a conjecture of Curien and Kortchemski.

Mots clés: Random planar maps; Scaling Limits; Local limits; O(n) loop model

Domaines: Probabilités [math.PR]

Loïc Richier. Limits of the boundary of random planar maps. 2017. ⟨hal-01503295v2⟩