On large $3/2$-stable maps

Abstract

We discuss asymptotics of large Boltzmann random planar maps such that every vertex of degree $k$ has weight of order $k^{-2}$. Infinite maps of that kind were studied by Budd, Curien and Marzouk. These maps can be seen as the dual of the discrete $α$-stable maps studied by Le Gall and Miermont for $α=3/2$ or as the gaskets of critical $O(2)$-decorated random planar maps. We compute the asymptotics of the graph distance and of the first passage percolation distance between two uniform vertices, which are respectively equivalent in probability to $(\log \ell)^2/π^2$ and $2(\log \ell)/(π^2 p_{\bf q})$ when the perimeter of the map $\ell$ goes to $\infty$, where $p_{\bf q}$ is a constant which depends on the model. We also show that the diameter is of the same order as those distances for both metrics and obtain in particular that these maps do not satisfy scaling limits in the sense of Gromov-Prokhorov or Gromov-Hausdorff for lack of tightness. To study the peeling exploration of these maps, we prove local limit and scaling limit theorems for a class of random walks with heavy tails conditioned to remain positive until they die at $-\ell$ towards processes that we call stable Lévy processes conditioned to stay positive until they jump and die at $-1$.

Figures (6)

• Figure 1: Simulation of a $3/2$-stable map $\mathfrak{M}^{(500,\dagger)}$ of perimeter $1000$. The two large red spheres are two uniform random vertices of $\mathfrak{M}^{(500,\dagger)}$ and the red path is the shortest path between them.
• Figure 2: Simulation of the conditioned random walks $P_\infty$ and $P_{\ell}$ under the coupling $\mathbb{P}^{(1),n}_{\ell,\infty}$ in the case $\tau_{-\ell}>n$ for $n=1000$ and $\ell=2000$ (after time $n$, we let both processes evolve independently).
• Figure 3: A filled-in exploration of a map with a target face $\mathfrak{m}_\square \in \mathcal{M}^5_1$. The peeled edge is in red, the root face is in white, the target face is in black and the hole containing the target is in grey. At the first step, an event $C_4$ happens, then an event $G_{1,*}$ occurs and finally the event $C_1^\mathrm{stop}$ ends the exploration.
• Figure 4: From left to right: a map with a distinguished edge in red, the same map with the edge "unzipped" giving rise to a distinguished $2$-face (in grey) and finally the same map which has been re-rooted.
• Figure 5: Illustration of the proof of Theorem \ref{['distance entre deux arêtes uniformes']}: the map $\mathfrak{M}^{(\ell)}$ (or $\mathfrak{M}^{(\ell)}_1$) is represented as a cactus where the height corresponds to the distance to the root face $f_r$. The blue holes are the two holes of the explored region $\mathfrak{e}$. The target $2$-face $\square$ and the uniform random edge $E^{(\ell)}$ on $\mathfrak{M}^{(\ell)}_1$, in blue, correspond respectively to the random uniform edges $E_1^{(\ell)}$ and $E_2^{(\ell)}$ in $\mathfrak{M}^{(\ell)}$, in red.

Theorems & Definitions (60)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Theorem 2.1
• Theorem 2.2
• Lemma 2.3
• Proposition 2.4
• Lemma 2.5
• proof
• Proposition 2.6