The conformally invariant measure on self-avoiding loops

TL;DR

This paper introduces a unique conformally invariant measure on simple loops in the plane and Riemann surfaces, exploring its properties and implications for critical percolation clusters and Brownian loops.

Contribution

It establishes the existence and uniqueness of a natural conformally invariant measure on simple loops, with invariance under restriction, and examines its properties and applications.

Findings

Unique measure on simple loops established

Invariance under conformal transformations and restriction proven

Implications for percolation clusters and Brownian loops analyzed

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 surface S' that is contained in another Riemann surface S, is just the measure on S restricted to those loops that stay in S'). We study some of its properties and consequences concerning outer boundaries of critical percolation clusters and Brownian loops.