  We discuss duality properties of critical Boltzmann planar maps such that the
degree of a typical face is in the domain of attraction of a stable
distribution with parameter $\alpha\in(1,2]$. We consider the critical
Bernoulli bond percolation model on a Boltzmann map in the dilute and generic
regimes $\alpha \in (3/2,2]$, and show that the open percolation cluster of the
origin is itself a Boltzmann map in the dense regime $\alpha \in (1,3/2)$, with
parameter \[\alpha':= \frac{2\alpha+3}{4\alpha-2}.\] This is the counterpart in
random planar maps of the duality property $\kappa \leftrightarrow 16/\kappa$
of Schramm-Loewner Evolutions and Conformal Loop Ensembles, recently
established by Miller, Sheffield and Werner. As a byproduct, we identify the
scaling limit of the boundary of the percolation cluster conditioned to have a
large perimeter. The cases of subcritical and supercritical percolation are
also discussed. In particular, we establish the sharpness of the phase
transition through the tail distribution of the size of the percolation
cluster.
