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The bulk one-arm exponent for the CLE$_{κ'}$ percolations

Haoyu Liu, Xin Sun, Pu Yu, Zijie Zhuang

Abstract

The conformal loop ensemble (CLE) is a conformally invariant random collection of loops. In the non-simple regime $κ'\in (4,8)$, it describes the scaling limit of the critical Fortuin-Kasteleyn (FK) percolations. CLE percolations were introduced by Miller-Sheffield-Werner (2017). The CLE$_{κ'}$ percolations describe the scaling limit of a natural variant of the FK percolation called the fuzzy Potts model, which has an additional percolation parameter $r$. Based on CLE percolations and assuming that the convergence of the FK percolation to CLE, K{ö}hler-Schindler and Lehmkuehler (2022) derived all the arm exponents for the fuzzy Potts model except the bulk one-arm exponent. In this paper, we exactly solve this exponent, which prescribes the dimension of the clusters in CLE$_{κ'}$ percolations. As a special case, the bichromatic one-arm exponent for the critical 3-state Potts model should be $4/135$. To the best of our knowledge, this natural exponent was not predicted in physics. Our derivation relies on the iterative construction of CLE percolations from the boundary conformal loop ensemble (BCLE), and the coupling between Liouville quantum gravity and SLE curves. The source of the exact solvability comes from the structure constants of boundary Liouville conformal field theory. A key technical step is to prove a conformal welding result for the target-invariant radial SLE curves. As intermediate steps in our derivation, we obtain several exact results for BCLE in both the simple and non-simple regimes, which extend results of Ang-Sun-Yu-Zhuang (2024) on the touching probability of non-simple CLE. This also provides an alternative derivation of the relation between the BCLE parameter $ρ$ and the additional percolation parameter $r$ in CLE percolations, which was originally due to Miller-Sheffield-Werner (2021, 2022).

The bulk one-arm exponent for the CLE$_{κ'}$ percolations

Abstract

The conformal loop ensemble (CLE) is a conformally invariant random collection of loops. In the non-simple regime , it describes the scaling limit of the critical Fortuin-Kasteleyn (FK) percolations. CLE percolations were introduced by Miller-Sheffield-Werner (2017). The CLE percolations describe the scaling limit of a natural variant of the FK percolation called the fuzzy Potts model, which has an additional percolation parameter . Based on CLE percolations and assuming that the convergence of the FK percolation to CLE, K{ö}hler-Schindler and Lehmkuehler (2022) derived all the arm exponents for the fuzzy Potts model except the bulk one-arm exponent. In this paper, we exactly solve this exponent, which prescribes the dimension of the clusters in CLE percolations. As a special case, the bichromatic one-arm exponent for the critical 3-state Potts model should be . To the best of our knowledge, this natural exponent was not predicted in physics. Our derivation relies on the iterative construction of CLE percolations from the boundary conformal loop ensemble (BCLE), and the coupling between Liouville quantum gravity and SLE curves. The source of the exact solvability comes from the structure constants of boundary Liouville conformal field theory. A key technical step is to prove a conformal welding result for the target-invariant radial SLE curves. As intermediate steps in our derivation, we obtain several exact results for BCLE in both the simple and non-simple regimes, which extend results of Ang-Sun-Yu-Zhuang (2024) on the touching probability of non-simple CLE. This also provides an alternative derivation of the relation between the BCLE parameter and the additional percolation parameter in CLE percolations, which was originally due to Miller-Sheffield-Werner (2021, 2022).

Paper Structure

This paper contains 22 sections, 44 theorems, 168 equations, 13 figures.

Table of Contents

  1. Introduction
  2. One-arm exponent for the CLE percolations and the fuzzy Potts model
  3. Boundary CLE: winding orientation and conformal radius distribution
  4. Proof of Theorems \ref{['thm:touch-BCLE-simple']}--\ref{['thm:CR-BCLE-nonsimple']} based on Liouville quantum gravity
  5. Outlook and perspectives
  6. CLE percolations, BCLE, and the fuzzy Potts model
  7. Description of CLE percolations via BCLE
  8. Derivation of the one-arm exponent given Theorems \ref{['thm:CR-BCLE-simple']} and \ref{['thm:CR-BCLE-nonsimple']}
  9. Fuzzy Potts one-arm exponent
  10. Bichromatic one-arm exponent for 3-states Potts model
  11. Liouville quantum gravity surfaces
  12. Liouville fields and quantum surfaces
  13. Pinched thin quantum annulus
  14. Simple BCLE loop from conformal welding
  15. Conformal welding of quantum surfaces
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $\kappa=16/\kappa' \in (2,4)$. The bulk one-arm exponent $\alpha_1(r)$ exists and is given by the unique positive solution in $(0,1-\frac{2}{\kappa'}-\frac{3\kappa'}{32})$ to the equation where $\rho = \frac{2}{\pi} \arctan (\frac{\sin(\pi \kappa/2)}{1 + \cos(\pi \kappa/2) - 1/(1-r)} ) - 2 \in (-2, \kappa-4)$.

Figures (13)

  • Figure 1: Left: A critical 3-state Potts configuration in the box $\Lambda_n$ drawn on the dual lattice, where vertices assigned with spin $1,2,3$ are colored in red, yellow and green, respectively. If we treat yellow and green vertices as the same color (say, blue), it becomes the fuzzy Potts configuration with $q=3$ and $r=1/3$. Shown in solid (resp. dashed) black are the boundary-touching (resp. non-boundary touching) red/blue interfaces, and the former should converge to $\mathrm{BCLE}_{10/3}(-1)$. Middle: The green one-arm event inside $\Lambda_{m,n}$ requests the existence of a green path from $\partial \Lambda_m$ to $\partial \Lambda_n$. The one-arm exponent is equal to $7/80$. Right. The bichromatic one-arm event inside $\Lambda_{m,n}$ requests the existence of a path consisting of green or yellow vertices. Corollary \ref{['cor:3-Potts-bichromatic']} shows that the bichromatic one-arm exponent is $4/135$.
  • Figure 2: Left: A thin quantum disk of weight $W<\frac{\gamma^2}{2}$. Note that the figure is not accurate since there are infinite number of beads near the two red marked points at the end and between each two beads. Right: A quantum triangle with $W_2\geq\frac{\gamma^2}{2}$ and $W_1,W_3<\frac{\gamma^2}{2}$ embedded as $(D,\phi,a_1,a_2,a_3).$ The two thin disks (colored green) are concatenated with the thick triangle (colored yellow) at points $\tilde{a}_1$ and $\tilde{a}_3$.
  • Figure 3: An illustration of pinched quantum annulus of weight $W$. In Theorem \ref{['thm:weld-BCLE']} we prove that when glued to the blue disk from $\mathrm{QD}_{1,0}$, we get another sample from $\mathrm{QD}_{1,0}$ decorated with an independent (counterclockwise) BCLE loop $\mathcal{L}$.
  • Figure 4: Left: Illustration of Theorem \ref{['thm:disk-welding']} with $W_1\ge\frac{\gamma^2}{2}$ and $W_2<\frac{\gamma^2}{2}$. Right: Illustration of Theorem \ref{['thm:disk+QT']} with $W,W_3\ge\frac{\gamma^2}{2}$ and $W_1,W_2<\frac{\gamma^2}{2}$.
  • Figure 5: An illustration of Lemma \ref{['lem:BCLEloop-0']}. Left: The loop $\mathcal{L}$ is a counterclockwise loop, and the branch $\eta^w$ is the union of the blue and the red curve. In this setting $z_0\in I_\eta$ and $z_1 = w$. Right: An illustration of the branch $\eta^{z_1}$. 0 is not surrounded by $\eta^{z_1}$ and $z_0$ is not on the arc $I_\eta$.
  • ...and 8 more figures

Theorems & Definitions (87)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.10
  • Theorem 1.11
  • Theorem 2.1
  • ...and 77 more