Many "local" models of random planar maps, obtained by gluing uniformly at random the edges of a collection of polygons with given degree, in order to obtain a topological sphere, are known to converge to the so-called Brownian sphere, a canonical model of random surface. However, it is possible to escape from the wide universality class of the Brownian sphere, either by considering models of random planar maps with long-range correlations, typically by endowing them with a statistical physics model at criticality, or by considering local models in which the variance of the face degrees distribution is infinite. In this talk, we investigate this second question by showing that random Boltzmann maps whose face degree distributions belong to a stable domain of attraction converge in the scaling limit to a random fractal object, which is called a stable carpet or a stable gasket depending on the value of the stable exponent. Indeed, a phase transition for the topology of these objects occur at the value 3/2 of the stable exponent. This completes earlier results by Le Gall-Miermont, who obtained a scaling limit result only up to extraction of subsequences. This is joint work with Nicolas Curien and Armand Riera.