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Limits of the boundary of random planar maps

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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 \(1/(\alpha -1/2)\). 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.

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Acknowledgements

Many thanks to Grégory Miermont for enlightening discussions and for attentively reading this work. I would also like to thank warmly Erich Baur, Jérémie Bouttier, Timothy Budd, Nicolas Curien and Igor Kortchemski for very useful discussions and comments. Finally, I am deeply indebted to the anonymous referees for their careful rereading, crucial corrections and remarks.

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  1. CMAP, École polytechnique, Palaiseau, France

    Loïc Richier

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  1. Loïc Richier
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Correspondence to Loïc Richier.

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This work was partially accomplished at UMPA, École Normale Supérieure de Lyon and supported by the Grant ANR-14-CE25-0014 (ANR GRAAL).

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Richier, L. Limits of the boundary of random planar maps. Probab. Theory Relat. Fields 172, 789–827 (2018). https://doi.org/10.1007/s00440-017-0820-y

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