Up-to-constants estimates on four-arm events for simple conformal loop ensemble

TL;DR

The paper derives up-to-constants estimates for general four-arm events in simple CLE$_{\kappa}$, with $\kappa\in(8/3,4]$, by leveraging the Brownian loop soup representation and a cluster-type separation lemma to handle all topologies of arm crossings. The main results yield explicit exponents: $\xi^{+}_{4}(\kappa)=\dfrac{2(12-\kappa)}{\kappa}$ for boundary arms and $\xi_{4}(\kappa)=\dfrac{(12-\kappa)(\kappa+4)}{8\kappa}$ for interior arms, with corresponding SLE extensions $\mathbb{P}[\mathcal{W}^{+}_4]\asymp\varepsilon^{\xi^{+}_{4}(\kappa)}$ and $\mathbb{P}[\mathcal{W}_4]\asymp\varepsilon^{\xi_{4}(\kappa)}$. The work develops a robust separation framework, proves a strengthened interior separation lemma, and demonstrates equivalence of multiple four-arm event definitions, enabling a comprehensive arm-exponent program for both CLE and SLE in this regime. These results provide crucial input for percolation-type questions related to level sets in discrete Gaussian free fields and loop-soup occupation fields, connecting continuum CLE/SLE exponents to discrete models. Overall, the paper closes gaps in prior analyses and establishes a versatile toolkit for arm-event estimates in the simple CLE/SLE setting.

Abstract

We prove up-to-constants estimates for a general class of four-arm events in simple conformal loop ensembles, i.e. CLE$_κ$ for $κ\in (8/3,4]$. The four-arm events that we consider can be created by either one or two loops, with no constraint on the topology of the crossings. Our result is a key input in our series of works arxiv:2409.16230 and arxiv:2409.16273 on percolation of the two-sided level sets in the discrete Gaussian free field (and level sets in the occupation field of the random walk loop soup). In order to get rid of all constraints on the topology of the crossings, we rely on the Brownian loop-soup representation of simple CLE [Ann. Math. 176 (2012) 1827-1917], and a "cluster version" of a separation lemma for the Brownian loop soup. As a corollary, we also obtain up-to-constants estimates for a general version of four-arm events for SLE$_κ$ for $κ\in (8/3,4]$. This fixes (in the case of four arms and $κ\in(8/3,4]$) an essential gap in [Ann. Probab. 46 (2018) 2863-2907] and improves some estimates therein.

Figures (12)

• Figure 1.1: We depict three possible configurations of $\mathcal{A}_4(\varepsilon, r)$. The left and middle figures belong to the event $\mathcal{E}_1(\varepsilon, r)$. The right figure belongs to the event $\mathcal{E}_2(\varepsilon, r)$.
• Figure 1.2: The well-separated arm event $\widetilde{\mathcal{E}_2}(\varepsilon, r)$.
• Figure 1.3: We depict three simple cases of boundary four-arm events (the two half-disks, centered on $x$ and $y$, have respective radii $\varepsilon$ and $r$). None of them belongs to the event $\mathcal{H}^\alpha_3(\varepsilon, x, y,r)$ or $\mathcal{H}^\alpha_4(\varepsilon, x, y,r)$ considered in MR3846840: roughly speaking, the curve does not hit the right arcs in the right order (see \ref{['eq:h1']} and \ref{['eq:h2']} for precise definitions). Our Theorem \ref{['thm:sle_int_utc']} implies in particular that their probabilities are upper-bounded by a constant times $\varepsilon^{\xi^+_4(\kappa)}$.
• Figure 1.4: We depict three simple cases of interior four-arm events (where the disk around $z$ and the half-disk around $y$ have radii $\varepsilon$ and $r$, respectively). None of them belongs to the event $\mathcal{E}_4(\varepsilon, z, y,r)$ considered in MR3846840 (see \ref{['eq:int_arm_event']} for a precise definition). Our Theorem \ref{['thm:sle_int_utc']} implies, after a conformal map from $\mathbb{H}$ onto $\mathbb{D}$, that the probabilities of these events are upper-bounded by a constant times $\varepsilon^{\xi_4(\kappa)}$.
• Figure 3.1: Exploration process \ref{['expl1']}. The red curve is $\ell([0,t_1])$. The brown curve is $\widehat{\gamma}$.

Theorems & Definitions (49)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1: Koebe $1/4$ in the upper half-plane
• proof
• Proposition 2.2: Corollary 5, MR2435854
• Proposition 2.3: MR2574734
• Theorem 2.4: Theorem 1.1, MR4082185
• Theorem 2.5: Theorem 1.1, MR4235483
• Lemma 2.6