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 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.

v3: minor changes, to appear in the Journal of the AMS

References in corpus (3)

• SLE martingales and the Virasoro algebra
• The Full Scaling Limit of Two-Dimensional Critical Percolation
• On Chordal and Bilateral SLE in multiply connected domains

Cited by in corpus (5)

• The expected area of the filled planar Brownian loop is Pi/5
• SLE(6) and CLE(6) from Critical Percolation
• Correlation functions of twist operators applied to single self-avoiding loops
• Critical exponents for the homology of Fortuin-Kasteleyn clusters on a torus
• Geometric Exponents of Dilute Logarithmic Minimal Models