The scaling limit of planar maps with large faces

Abstract

We prove that large Boltzmann stable planar maps of index $\alpha \in (1;2)$ converge in the scaling limit towards a random compact metric space ${\mathcal{S}}_{\alpha }$ 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 $\alpha \in [3/2;2)$, the topology of ${\mathcal{S}}_{\alpha }$ is that of the Sierpinski carpet, while in the dense phase $\alpha \in (1;3/2)$ the ``faces'' of ${\mathcal{S}}_{\alpha }$ may touch each-others. En route, we prove various geometric properties of these objects concerning their faces or the behavior of geodesics.

arXiv comment187 pages, 47 figures. Comments are welcome!