A simple construction of Werner measure from chordal SLE$_{8/3}$

TL;DR

This paper constructs the Werner measure on self-avoiding loops using chordal SLE_{8/3}, providing new proofs, covariance properties, and explicit bounds for the measure, advancing understanding of conformally invariant loop measures.

Contribution

It offers a direct construction of the Werner measure from SLE_{8/3} and establishes its uniqueness and properties through new proofs and explicit calculations.

Findings

Constructed Werner measure from chordal SLE_{8/3}.

Proved uniqueness of the measure.

Derived explicit bounds for the loop measure.

Abstract

We give a direct construction of the conformally invariant measure on self-avoiding loops in Riemann surfaces (Werner measure) from chordal $\text{SLE}_{8/3}$. We give a new proof of uniqueness of the measure and use Schramm's formula to construct a measure on boundary bubbles encircling an interior point. After establishing covariance properties for this bubble measure, we apply these properties to obtain a measure on loops by integrating measures on boundary bubbles. We calculate the distribution of the conformal radius of boundary bubbles encircling an interior point and deduce from it explicit upper and lower bounds for the loop measure.