  We study the geometry of infinite random Boltzmann planar maps with vertices
of high degree. These correspond to the duals of the Boltzmann maps associated
to a critical weight sequence $(q_{k})_{ k \geq 0}$ for the faces with
polynomial decay $k^{-a}$ with $a \in ( 3/2, 5/2)$ which have been studied by
Le Gall & Miermont as well as by Borot, Bouttier & Guitter. We show the
existence of a phase transition for the geometry of these maps at $a = 2$. In
the dilute phase corresponding to $a \in (2, 5/2)$ we prove that the volume of
the ball of radius $r$ (for the graph distance) is of order $r^{\mathsf{d}}$
with $\mathsf{d}= (a-1/2)/(a-2)$, and we provide distributional scaling limits
for the volume and perimeter process. In the dense phase corresponding to $a
\in (3/2,2)$ the volume of the ball of radius $r$ is exponential in $r$. We
also study the first-passage percolation (FPP) distance with exponential edge
weights and show in particular that in the dense phase the FPP distance between
the origin and infinity is finite. The latter implies in addition that the
random lattices in the dense phase are transient. The proofs rely on the recent
peeling process introduced in arXiv:1506.01590 and use ideas of arXiv:1412.5509
in the dilute phase.
