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A note on Loewner energy, conformal restriction and Werner's measure on self-avoiding loops

Yilin Wang

Abstract

In this note, we establish an expression of the Loewner energy of a Jordan curve on the Riemann sphere in terms of Werner's measure on simple loops of SLE$_{8/3}$ type. The proof is based on a formula for the change of the Loewner energy under a conformal map that is reminiscent of the restriction properties derived for SLE processes.

A note on Loewner energy, conformal restriction and Werner's measure on self-avoiding loops

Abstract

In this note, we establish an expression of the Loewner energy of a Jordan curve on the Riemann sphere in terms of Werner's measure on simple loops of SLE type. The proof is based on a formula for the change of the Loewner energy under a conformal map that is reminiscent of the restriction properties derived for SLE processes.

Paper Structure

This paper contains 5 sections, 7 theorems, 48 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Werner's measure on self-avoiding loops
  3. Conformal restriction for simple chord
  4. Conformal restriction for simple loop
  5. Loewner energy as a renormalization of Werner's measure

Key Result

Theorem 1

If $g = e^{2\sigma} g_0$ is conformally equivalent to the spherical metric $g_0$ on $S^2$, then

Figures (3)

  • Figure 1: Maps in the proof of Proposition \ref{['prop_energy_change']}, $\tilde{W}_t = \psi_t(W_t)$.
  • Figure 2: Maps in the proof of Theorem \ref{['thm_restr_loop']}.
  • Figure 3: Maps in the proof of Lemma \ref{['lem_anal']}.

Theorems & Definitions (13)

  • Theorem 1: W2 Proposition 7.1, Theorem 7.3
  • Proposition 3.1: W1 Proposition 4.1
  • proof
  • Corollary 3.2
  • Theorem 4.1
  • Remark
  • proof
  • Lemma 4.2
  • proof
  • Corollary 4.3
  • ...and 3 more