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From the random geometry of conformally invariant systems to the Kähler geometry of universal Teichmüller space

Yilin Wang

Abstract

The goal of this expository article is to explain how a fundamental functional on the space of Jordan curves arising from SLE - Loewner energy - is connected to a seemingly far apart subject: the Kähler geometry of universal Teichmüller space. We provide a background on Loewner energy, the universal Liouville action, and the intuition behind the proof of the identity between them.

From the random geometry of conformally invariant systems to the Kähler geometry of universal Teichmüller space

Abstract

The goal of this expository article is to explain how a fundamental functional on the space of Jordan curves arising from SLE - Loewner energy - is connected to a seemingly far apart subject: the Kähler geometry of universal Teichmüller space. We provide a background on Loewner energy, the universal Liouville action, and the intuition behind the proof of the identity between them.
Paper Structure (8 sections, 7 theorems, 29 equations, 2 figures)

This paper contains 8 sections, 7 theorems, 29 equations, 2 figures.

Table of Contents

  1. Conformally invariant simple random curves in the plane
  2. Loewner energy and SLE
  3. Identity with the Universal Liouville action
  4. Weil--Petersson Teichmüller space
  5. How did we come up with the identity?
  6. Outline of the proof of Theorem thm:main
  7. Further discussions
  8. Acknowledgment

Key Result

Theorem 2

Let $\Omega$(resp., $\Omega^*$) denote the component of $\hat{\mathbb{C}} \smallsetminus \gamma$ which does not contain $\infty$(resp., which contains $\infty$) and $f$(resp., $g$) be a conformal map from the unit disk $\mathbb{D} = \{z \in \hat{\mathbb{C}} \colon |z| < 1 \}$ onto $\Omega$(resp., fr where $g'(\infty):=\lim_{z\to \infty} g'(z)$ and $\mathrm{d}^2 z$ is the Euclidean area measure.

Figures (2)

  • Figure 1: The left arrow illustrates a scaling map, and the right arrow illustrates the uniformizing conformal map from the slit domain onto the upper half-plane. The law of the random chord should be identical in these three pictures.
  • Figure 2: An illustration of the rare event of $\sqrt \kappa B$ being close to a deterministic function $W$ (whose graph is the solid red line). The blue curve is a simulation of $\sqrt \kappa B$ with $\kappa = 0.1$ over $10000$ steps conditioned to stay close to $W$.

Theorems & Definitions (9)

  • Definition 1
  • Theorem 2: See W2
  • Corollary 3
  • Theorem 4: See BowickRajeev1987string
  • proof
  • Theorem 5: See TT06
  • Theorem 6: See TT06
  • Theorem 7: See W2
  • Theorem 8: See W2