# SLE Loop Measures

## TL;DR

This paper constructs various SLE$_ppa$ loop and bubble measures using Minkowski content, establishing their conformal properties and answering open questions about their invariance and homogeneity.

## Contribution

It introduces new SLE$_ppa$ loop measures with conformal invariance, reversibility, and space-time homogeneity, extending previous theories and addressing open questions.

## Key findings

- Constructed rooted SLE$_ppa$ loop measures with conformal properties.
- Extended loop measures to subdomains and Riemann surfaces using Brownian loop measures.
- Provided examples of loop measures for all $c \u2264 1$ and answered a question on space-time homogeneity.

## Abstract

We use Minkowski content (i.e., natural parametrization) of SLE to construct several types of SLE$_\kappa$ loop measures for $\kappa\in(0,8)$. First, we construct rooted SLE$_\kappa$ loop measures in the Riemann sphere $\widehat{\mathbb C}$, which satisfy M\"obius covariance, conformal Markov property, reversibility, and space-time homogeneity, when the loop is parametrized by its $(1+\frac \kappa 8)$-dimensional Minkowski content. Second, by integrating rooted SLE$_\kappa$ loop measures, we construct the unrooted SLE$_\kappa$ loop measure in $\widehat{\mathbb C}$, which satisfies M\"obius invariance and reversibility. Third, we extend the SLE$_\kappa$ loop measures from $\widehat{\mathbb C}$ to subdomains of $\widehat{\mathbb C}$ and to two types of Riemann surfaces using Brownian loop measures, and obtain conformal invariance or covariance of these measures. Finally, using a similar approach, we construct SLE$_\kappa$ bubble measures in simply/multiply connected domains rooted at a boundary point. The SLE$_\kappa$ loop measures for $\kappa\in(0,4]$ give examples of Malliavin-Kontsevich-Suhov loop measures for all $c\le 1$. The space-time homogeneity of rooted SLE$_\kappa$ loop measures in $\widehat{\mathbb C}$ answers a question raised by Greg Lawler.