The scaling limit of planar maps with large faces

TL;DR

This work identifies the scaling limit of large, non-generic Boltzmann planar maps with exponent $α\in(1,2)$ as a universal random compact metric space, the $α$-stable carpet/gasket $\mathcal{S}$, constructed from an $α$-stable Lévy excursion via an $α$-stable looptree and a Gaussian label field. The authors develop a continuum theory based on the BDG encoding, stable looptrees, and a Gaussian Free Field-like label process, then connect it to discrete Boltzmann maps through BDG encodings to prove convergence in the Gromov–Hausdorff–Prokhorov topology. They demonstrate a phase transition: in the dilute regime ($α∈[3/2,2)$) the limiting topology is the Sierpiński carpet, while in the dense regime ($α∈(1,3/2)$) faces may touch, and they establish detailed geometric and geodesic properties, including a sharp two-point function and a robust ball-volume control. The results illuminate strong links between stable random geometry, Brownian geometry, and Liouville quantum gravity, and lay groundwork for further bridges to CLE/LQG in the stable setting. Overall, the paper provides a comprehensive framework combining probabilistic, combinatorial, and geometric techniques to understand the scaling limits of complex planar-map models beyond the Brownian sphere.

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.

Figures (47)

• Figure 1: Simulations of large non-generic critical random Boltzmann planar maps of index $\alpha \in \{1.9, 1.8, 1.7, 1.6, 1.5, 1.4, 1.3\}$ from top left to bottom right.
• Figure 2: From top left to bottom right: The stable excursion $X$, the looptree $\mathcal{L}$ coded by $X$ with colors indicating the loops, the label process $Z$, and finally the same looptree with colors indicating the values of the process $Z$.
• Figure 3: Illustration of the geometric underlying idea for the proof of $D=0\iff D^*=0$. The "faces" of $\mathcal{S}$ are represented by the blue "holes". Except for the trivial identifications, the distance $D$ cannot identify more points since they must be separated by two faces.
• Figure 4: Illustration of the neighborhood of a good point $x \in \gamma_{1,2}$. The geodesic $\gamma_{1,2}$ is drawn in red and blue. The faces $\mathfrak{F}_1$ and $\mathfrak{F}_2$ are drawn in light blue and green. The points $\rho_1$ and $\rho_2$ could be in the same connected component as $x$ -- even if we will see that this is not the standard configuration.
• Figure 5: A simulation of a $\tfrac{3}{2}$-stable Lévy excursion and the corresponding looptree. Points belonging to the same loop (corresponding to the jumps of the excursion) are displayed with the same color.

Theorems & Definitions (145)

• Theorem 1.1: Scaling limit for non-generic Boltzmann maps
• Remark 1.2: Dual maps
• Theorem 1.3: Topology in the dilute case
• Proposition 2.1: Equivalence classes induced by $d$
• proof
• Proposition 2.2: Degree of points
• proof
• Remark 2.3: Hausdorff dimensions and a heuristic
• Lemma 2.4
• proof