The scaling limit of planar maps with large faces

Abstract

We prove that large Boltzmann stable planar maps of index $α\in (1;2)$ converge in the scaling limit towards a random compact metric space $\mathcal{S}_α$ that we construct explicitly. They form a one-parameter family of random continuous spaces ``with holes'' or ``faces'' different from the Brownian sphere. In the so-called dilute phase $α\in [3/2;2)$, the topology of $\mathcal{S}_α$ is that of the Sierpinski carpet, while in the dense phase $α\in (1;3/2)$ the ``faces'' of $\mathcal{S}_α$ may touch each-others. En route, we prove various geometric properties of these objects concerning their faces or the behavior of geodesics.