A simple construction of Werner measure from chordal SLE8/3

Abstract

We give a direct construction of the conformally invariant measure on self-avoiding loops in Riemann surfaces (Werner measure) from chordal $\mathrm{SLE}_{8/3}$. 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.