Distances on the $\mathrm{CLE}_4$, critical Liouville quantum gravity and $3/2$-stable maps

TL;DR

The paper advances the connection between CLE_4, Liouville quantum gravity, and 3/2-stable maps by proving that CLE_4 explorations on a 2-LQG disk carve the unexplored region into independent quantum disks and by linking the quantum natural distance to a conformally invariant boundary distance via a Lamperti-type transform. It establishes κ ↑ 4 convergence results that rigorously connect CLE_4 with its SLE-based approximations and transfers Markov properties to the uniform CLE_4 exploration. It further derives the scaling limit of distances from the boundary in large 3/2-stable maps, identifying a Lamperti-transformed quantum distance as the limiting metric and connecting peeling dynamics to quantum disk geometry. The work has broad implications for the scaling limits of random planar maps, including loop-decorated and O(2) decorated variants, and provides a framework for analyzing metric properties of CLE_4 decorated LQG surfaces.

Abstract

The purpose of this article is threefold. First, we show that when one explores a conformal loop ensemble of parameter $κ=4$ ($\mathrm{CLE}_4$) on an independent $2$-Liouville quantum gravity ($2$-LQG) disk, the surfaces which are cut out are independent quantum disks. To achieve this, we rely on approximations of the explorations of a $\mathrm{CLE}_4$: we first approximate the $\mathrm{SLE}_4^{\langle μ\rangle}(-2)$ explorations for $μ\in \mathbb{R}$ using explorations of the $\mathrm{CLE}_κ$ as $κ\uparrow 4$ and then we approximate the uniform exploration by letting $μ\to \infty$. Second, we describe the relation between the so-called natural quantum distance and the conformally invariant distance to the boundary introduced by Werner and Wu. Third, we establish the scaling limit of the distances from the boundary to the large faces of $3/2$-stable maps and relate the limit to the $\mathrm{CLE}_4$-decorated $2$-LQG.

Figures (7)

• Figure 1: Left: illustration of the $\mathrm{SLE}^{\langle \mu \rangle}_4(-2)$ exploration of the $\mathrm{CLE}_4$ ensemble. Right: illustration of the uniform exploration of the $\mathrm{CLE}_4$ ensemble. The blue curve $\eta$ is the trunk of the exploration. The unexplored region is in green, the domains encircled by loops are in orange and the cut out domains are in blue.
• Figure 2: Illustration of a chordal $\mathrm{SLE}_\kappa^\beta(\kappa-6)$ or $\mathrm{SLE}^{\langle \mu \rangle}_4(-2)$ from $0$ to $\infty$ in $\mathbb{H}$. Above: at a time $u$ where $\gamma$ is drawing a loop, i.e. a time such that $O_u\neq W_u$. Below: at the time $v$ where $\gamma$ closes the loop, so that $O_v=W_v$.
• Figure 3: Illustration of the proof of the Carathéodory convergence of the domain encircled by $\mathcal{L}_n= g^{\kappa_n}_{S^{\kappa_n}_{i,\varepsilon}}(\gamma^{\kappa_n}([S^{\kappa_n}_{\varepsilon, i}, T^{\kappa_n}_{\varepsilon,i}]))$, in green dashes, towards the domain encircled by $\mathcal{L} = g^4_{S^4_{i,\varepsilon}}(\gamma^4([S^4_{\varepsilon, i}, T^4_{\varepsilon,i}]))$, in black. The complement of $V_\delta$ is in orange. A compact $\mathcal{K}$ encircled by $\mathcal{L}$ is drawn in blue.
• Figure 4: Left: the Brownian motion $(B_s)_{s\ge 0}$. Center: a radial exploration of the $\mathrm{CLE}_4$. The red and blue excursions of height at least $\varepsilon$ give rise to two loops which do not encircle zero and the green part of the Brownian motion generates a loop which encircles zero. Right: approximation of the radial exploration.
• Figure 5: Left: a map $\mathfrak{e}$ with one hole which is a submap of the map $\mathfrak{m}$ on the right. The map $\mathfrak{u}$ which is in the hole of $\mathfrak{e}$ in the centre enables to recover $\mathfrak{m}$ after gluing the boundary of $\mathfrak{u}$ to the boundary of the hole.

Theorems & Definitions (65)

• Remark 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Lemma 2.1
• Lemma 2.2
• Proposition 3.1
• Proposition 3.2
• Proposition 3.3
• Lemma 3.4