The conformally invariant measure on self-avoiding loops

Abstract

We show that there exists a unique (up to multiplication by constants) and natural measure on simple loops in the plane and on each Riemann surface, such that the measure is conformally invariant and also invariant under restriction (i.e. the measure on a Riemann surfaceS′S’that is contained in another Riemann surfaceSSis just the measure onSSrestricted to those loops that stay inS′S’). We study some of its properties and consequences concerning outer boundaries of critical percolation clusters and Brownian loops.