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

TL;DR

This paper links Loewner energy of Jordan curves to Werner's measure on SLE$_{8/3}$ loops, providing a new formula based on conformal map properties and restriction principles.

Contribution

It introduces a novel expression connecting Loewner energy with Werner's measure, enhancing understanding of conformal invariance in loop measures.

Findings

Loewner energy can be expressed via Werner's measure on simple loops.

The change in Loewner energy under conformal maps follows a restriction-like formula.

The results deepen the connection between Loewner energy and SLE loop measures.

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.