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Decomposition of Hypergeometric SLE and Reversibility

Mingchang Liu

Abstract

In this paper, we consider hypergeometric SLE process for $κ\in (4,8)$ and $ν>\fracκ{2}-6$. Though the definition of hypergeometric SLE process is complicated, we show that given its hitting point on a specific boundary, its conditional law can be described by SLE$_κ(\underlineρ)$ process. Based on this observation, by constructing a pair of curves, we derive the reversibility of hypergeometric SLE for $κ\in(4,8)$ and $ν>-2$.

Decomposition of Hypergeometric SLE and Reversibility

Abstract

In this paper, we consider hypergeometric SLE process for and . Though the definition of hypergeometric SLE process is complicated, we show that given its hitting point on a specific boundary, its conditional law can be described by SLE process. Based on this observation, by constructing a pair of curves, we derive the reversibility of hypergeometric SLE for and .
Paper Structure (8 sections, 8 theorems, 38 equations, 1 figure)

This paper contains 8 sections, 8 theorems, 38 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Curve space
  4. Loewner chain and $\mathop{\mathrm{SLE}}\nolimits$
  5. Hypergeometric $\mathop{\mathrm{SLE}}\nolimits$
  6. $\mathop{\mathrm{SLE}}\nolimits_\kappa(\underline\rho)$ process and Gaussian free field
  7. Proof of Theorem \ref{['thm::decom']}
  8. Proof of Theorem \ref{['thm::reversibility']}

Key Result

Theorem 1.1

Fix $\kappa\in (4,8)$ and $\nu>\frac{\kappa}{2}-6$. Let $\eta$ be the $\mathop{\mathrm{hSLE}}\nolimits_\kappa(\nu)$ curve in $(\mathbb{H};0,x,y,\infty)$ from $0$ to $\infty$ with force points $x$, $y$. We denote by $\tau$ the hitting time of $\eta$ at $(y,+\infty)$. Then, the density of $\eta(\tau)$ Moreover, the conditional distribution of $\eta[0,\tau]$ given $\eta(\tau)$ is $\mathop{\mathrm{SLE

Figures (1)

  • Figure 3.1: This is an illustration image under $f$. We require $f(y)=0$ and $f(\infty)$=1. The corresponding angles of the quad are shown in picture.

Theorems & Definitions (16)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • proof : Proof of Theorem \ref{['thm::decom']}
  • Corollary 3.3
  • proof
  • Lemma 4.1
  • ...and 6 more