  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.

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