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Large deviations of Schramm-Loewner evolutions: A survey

Yilin Wang

Abstract

These notes survey the first results on large deviations of Schramm-Loewner evolutions (SLE) with emphasis on interrelations between rate functions and applications to complex analysis. More precisely, we describe the large deviations of SLE$_κ$ when the $κ$ parameter goes to zero in the chordal and multichordal case and to infinity in the radial case. The rate functions, namely Loewner and Loewner-Kufarev energies, are closely related to the Weil-Petersson class of quasicircles and real rational functions.

Large deviations of Schramm-Loewner evolutions: A survey

Abstract

These notes survey the first results on large deviations of Schramm-Loewner evolutions (SLE) with emphasis on interrelations between rate functions and applications to complex analysis. More precisely, we describe the large deviations of SLE when the parameter goes to zero in the chordal and multichordal case and to infinity in the radial case. The rate functions, namely Loewner and Loewner-Kufarev energies, are closely related to the Weil-Petersson class of quasicircles and real rational functions.

Paper Structure

This paper contains 25 sections, 48 theorems, 137 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Large deviation principle
  3. Chordal Loewner chain
  4. Chordal SLE
  5. Large deviations of chordal $\operatorname{SLE}_{0+}$
  6. Chordal Loewner energy and large deviations
  7. Reversibility of Loewner energy
  8. Loop energy and Weil-Petersson quasicircles
  9. Cutting, welding, and flow-lines
  10. Cutting-welding identity
  11. Flow-line identity
  12. Applications
  13. Large deviations of multichordal SLE$_{0+}$
  14. Multichordal SLE
  15. Real rational functions and Shapiro's conjecture
  16. ...and 10 more sections

Key Result

Theorem 1.5

If $\mathcal{X}, \mathcal{Y}$ are two Polish spaces, $f: \mathcal{X} \to \mathcal{Y}$ a continuous function, and a family of probability measures $\{\mu_\varepsilon\}_{\varepsilon > 0}$ on $\mathcal{X}$ satisfying the large deviation principle with good rate function $I: \mathcal{X} \to [0,\infty]$. Then the family of pushforward probability measures $\{f_* \mu_\varepsilon\}_{\varepsilon > 0}$ on

Figures (3)

  • Figure 1: From chord in $(\mathbb{H}; 0, \infty)$ to a Jordan curve.
  • Figure 2: Illustration of a multichord and the domain $\hat{D}_j$ containing $\gamma_j$.
  • Figure 3: Illustration of the winding function $\varphi$.

Theorems & Definitions (115)

  • Definition 1.1
  • Definition 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.5: Contraction principle DZ10
  • Theorem 1.6
  • Remark 1.7
  • Definition 1.8
  • Example 1.9
  • Theorem 1.10: Dawson-Gärtner DZ10
  • ...and 105 more