Limits of the boundary of random planar maps

Abstract

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 $(\alpha -1/2{)}^{-1}$. 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.

34 pages