The scaling limit of planar maps with large faces

TL;DR

This paper establishes the scaling limits of large planar maps with faces governed by a stable distribution, revealing a family of fractal-like spaces with diverse topologies depending on the parameter.

Contribution

It constructs and characterizes a new family of random metric spaces as limits of large stable planar maps, extending understanding of geometric structures in random geometry.

Findings

Convergence of large stable planar maps to a family of fractal-like metric spaces.

Identification of the topology of limit spaces as Sierpinski carpet in the dilute phase.

Analysis of geometric properties such as faces and geodesic behavior.

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.