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1 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées

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 $\alpha\in1,3-2$, 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 $\alpha\in3-2,2$, 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 $On$ loop model on quadrangulations, proving thereby a conjecture of Curien and Kortchemski.

Keywords : Random planar maps Scaling Limits Local limits On loop model





Author: Loïc Richier -

Source: https://hal.archives-ouvertes.fr/



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