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Boundary four-point connectivities of conformal loop ensembles

Gefei Cai

Abstract

We derive the boundary four-point Green's functions for conformal loop ensembles (CLE) with $κ\in(4,8)$. Specializing to $κ=6$ and $κ=16/3$, we establish the exact formulas for the boundary four-point connectivities in critical Bernoulli percolation and the FK-Ising model conjectured by Gori-Viti (2017, 2018). In particular, we identify a logarithmic singularity in the critical FK-Ising model. Our approach also applies to the one-bulk and two-boundary connectivities of CLE, thereby extending the factorization formula of Beliaev-Izyurov (2012) to all $κ\in(4,8)$.

Boundary four-point connectivities of conformal loop ensembles

Abstract

We derive the boundary four-point Green's functions for conformal loop ensembles (CLE) with . Specializing to and , we establish the exact formulas for the boundary four-point connectivities in critical Bernoulli percolation and the FK-Ising model conjectured by Gori-Viti (2017, 2018). In particular, we identify a logarithmic singularity in the critical FK-Ising model. Our approach also applies to the one-bulk and two-boundary connectivities of CLE, thereby extending the factorization formula of Beliaev-Izyurov (2012) to all .

Paper Structure

This paper contains 18 sections, 30 theorems, 123 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Boundary four-point connectivities of Bernoulli percolation
  3. Boundary four-point connectivities of CLE
  4. Discussions
  5. SLE bubble measures and CLE boundary Green's functions
  6. SLE bubble measure
  7. Boundary-touching CLE loops and unrooted SLE bubble measure
  8. CLE boundary Green's functions
  9. Smoothness and second-order PDEs
  10. Fusion
  11. Identification of solutions
  12. Proof of Proposition \ref{['prop:identify-v0']}
  13. Proof of Theorems \ref{['thm:percolation']} and \ref{['thm:fk']}
  14. Solutions of \ref{['eq:ode']} for other special $\kappa$
  15. One-Bulk and two-boundary connectivities
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $V_0,V_2$ be defined as above. There exists a constant $C\in(0,\infty)$ such that Here, the constant $A:=\frac{8\sqrt{3}\,\pi \sin\!\left(\frac{2\pi}{9}\right)}{135\cos\!\left(\frac{5\pi}{18}\right)}\in(0,\infty)$.

Figures (1)

  • Figure 1: Illustration for the $\mathop{\mathrm{CLE}}\nolimits_\kappa$ exploration interface $\zeta$ of $\widehat{\Gamma}$. The segment $\zeta[\sigma,\sigma']$ is colored red, while $\zeta[0,\sigma']$ is in orange. Left: the event $E_{r_3,r_4}\setminus F_{r_3,r_4}$, and $\eta_{12}$ is colored blue. Right: the event $F_{r_3,r_4}$, and $\eta_{12}$ is the union of the blue and red curves.

Theorems & Definitions (58)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark 1.6
  • Theorem 1.7
  • Lemma 2.1: shef-werner-cle
  • Proposition 2.2
  • proof
  • ...and 48 more