Limits of the boundary of random planar maps

TL;DR

This paper investigates the asymptotic behavior of the boundary of critical Boltzmann planar maps, revealing phase transitions and identifying limits of large loops, with implications for the $O(n)$ loop model.

Contribution

It establishes the scaling limits of the boundary as stable looptrees and characterizes phase transitions between tree-like and half-plane boundary structures.

Findings

Boundary converges to stable looptree in dense phase

Existence of phase transition in boundary structure

Limits of large loops in the $O(n)$ model are identified

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.