An SLE$_2$ loop measure

TL;DR

This paper constructs a conformally covariant measure on simple loops on Riemann surfaces corresponding to SLE$_2$, extending previous work and providing a new example of a loop measure with non-zero central charge.

Contribution

It introduces a new family of simple loop measures on Riemann surfaces with conformal covariance, corresponding to SLE$_2$, extending prior planar annuli results to general surfaces.

Findings

Constructed a conformally covariant loop measure for SLE$_2$ on Riemann surfaces.

Extended the loop measure from planar annuli to general Riemann surfaces.

Provided an alternative construction and analyzed conformal covariance properties.

Abstract

There is an essentially unique way to associate to any Riemann surface a measure on its simple loops, such that the collection of measures satisfy a strong conformal invariance property. Wendelin Werner constructed these random simple loops as boundaries of Brownian loops, and so they correspond in the zoo of statistical mechanics models to central charge $0$, or Schramm-Loewner Evolution (SLE) parameter $\kappa=8/3$. The goal of this paper is to construct a family of measures on simple loops on Riemann surfaces that satisfies a conformal covariance property, and that would correspond to SLE parameter $\kappa=2$ (central charge $-2$). On planar annuli, this loop measure was already built by Adrien Kassel and Rick Kenyon. We will give an alternative construction of this loop measure on planar annuli, investigate its conformal covariance, and finally extend this measure to general Riemann surfaces. This gives an example of a Malliavin-Kontsevich-Suhov loop measure in non-zero central charge.