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 (qk)k≥0 for the faces with polynomial decay k^-ɑ with ɑ ∈ (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 ɑ ∈ (3/2, 2) weprovethatthevolumeoftheballof radius r (for the graph distance) is of order r^d with d = (ɑ 1/2)/(ɑ 2), and we provide distributional scaling limits for the volume and perimeter process. In the dense phase corresponding to a ∈ (3/2,2)thevolumeoftheballofradiusrisexponentialinr. 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 ∞ 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 [16] and use ideas of [22] in the dilute phase.