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Multiple Ising interfaces in annulus and $2N$-sided radial SLE

Yu Feng, Hao Wu, Lu Yang

Abstract

We consider critical planar Ising model in annulus with alternating boundary conditions on the outer boundary and free boundary conditions in the inner boundary. As the size of the inner hole goes to zero, the event that all interfaces get close to the inner hole before they meet each other is a rare event. We prove that the law of the collection of the interfaces conditional on this rare event converges in total variation distance to the so-called $2N$-sided radial SLE$_3$, introduced by~[HL21]. The proof relies crucially on an estimate for multiple chordal SLE. Suppose $(γ_1, \ldots, γ_N)$ is chordal $N$-SLE$_κ$ with $κ\in (0,4]$ in the unit disc, and we consider the probability that all $N$ curves get close to the origin. We prove that the limit $\lim_{r\to 0+}r^{-A_{2N}}\mathbb{P}[\mathrm{dist}(0,γ_j)<r, 1\le j\le N]$ exists, where $A_{2N}$ is the so-called $2N$-arm exponents and $\mathrm{dist}$ is Euclidean distance. We call the limit Green's function for chordal $N$-SLE$_κ$. This estimate is a generalization of previous conclusions with $N=1$ and $N=2$ proved in~[LR12, LR15] and~[Zha20] respectively.

Multiple Ising interfaces in annulus and $2N$-sided radial SLE

Abstract

We consider critical planar Ising model in annulus with alternating boundary conditions on the outer boundary and free boundary conditions in the inner boundary. As the size of the inner hole goes to zero, the event that all interfaces get close to the inner hole before they meet each other is a rare event. We prove that the law of the collection of the interfaces conditional on this rare event converges in total variation distance to the so-called -sided radial SLE, introduced by~[HL21]. The proof relies crucially on an estimate for multiple chordal SLE. Suppose is chordal -SLE with in the unit disc, and we consider the probability that all curves get close to the origin. We prove that the limit exists, where is the so-called -arm exponents and is Euclidean distance. We call the limit Green's function for chordal -SLE. This estimate is a generalization of previous conclusions with and proved in~[LR12, LR15] and~[Zha20] respectively.
Paper Structure (19 sections, 21 theorems, 137 equations, 1 figure)

This paper contains 19 sections, 21 theorems, 137 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Convergence of Ising interfaces to radial SLE
  3. Green's function for multiple chordal SLE with $\kappa\in (0,4]$
  4. Multiple chordal SLE and multiple radial SLE
  5. Preliminaries: chordal $N$-$\mathrm{SLE}_{\kappa}$
  6. Preliminaries: $2N$-sided radial $\mathrm{SLE}$
  7. Transition density for chordal $N$-$\mathrm{SLE}_{\kappa}$
  8. Proof of Proposition \ref{['prop::multiplechordalestimates']}, Theorem \ref{['thm::NChordalSLEGreenFunction']} and Corollary \ref{['cor::chordaltoradial']}
  9. Ising model
  10. Convergence of polygons
  11. Convergence of annular polygons
  12. Convergence of multiple Ising interfaces in polygons
  13. Convergence of multiple Ising interfaces in annulus
  14. Convergence of Ising interfaces in polygons with one free segment
  15. Convergence of Ising interfaces in annulus
  16. ...and 4 more sections

Key Result

Theorem 1.1

Fix $\theta^1<\cdots<\theta^{2N}<\theta^1+\pi$ and write $\boldsymbol{\theta}=(\theta^1, \ldots, \theta^{2N})$. Denote $x_j=\exp(2\mathfrak{i}\theta^j)$ for $1\leq j\leq 2N$ and write $\boldsymbol{x}=(x_1, \ldots, x_{2N})$. For $p>0$, denote by $\mathbb{P}_{\mathrm{Ising}}^{(\mathbb{A}_p; \boldsymbo

Figures (1)

  • Figure 1.1: Consider critical Ising interfaces in annular polygon $(\mathbb{A}_p; x_1, x_2, x_3, x_4)$ with boundary conditions: $\oplus$ on $(x_1x_2)\cup(x_3x_4)$ and $\ominus$ on $(x_2x_3)\cup(x_4x_1)$ and free on the inner hole. There are four interfaces starting from $x_1, x_2, x_3, x_4$ respectively. There are seven possibilities for the connectivity of interfaces: in (a) and (b), four interfaces meet each other; in (c)-(f), two interfaces meet each other and the other two interfaces connect the marked points to the inner hole; in (g), all four interfaces hit the inner hole before they meet each other. In this article, we focus on the scenario in (g) and show that, conditional on this rare event, the law of the collection of the four interfaces converges to 4-sided radial $\mathrm{SLE}_3$.

Theorems & Definitions (42)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 1.3
  • Corollary 1.4
  • Proposition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • proof : Proof of Lemma \ref{['lem::RN_Palpha_P']}
  • ...and 32 more