SLE loop measures

Abstract

We use Minkowski content (i.e., natural parametrization) of SLE to construct several types of $$\hbox {SLE}_\kappa $$ loop measures for $$\kappa \in (0,8)$$ . First, we construct rooted $$\hbox {SLE}_\kappa $$ loop measures in the Riemann sphere $$\widehat{\mathbb {C}}$$ , which satisfy Mobius 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 $$\hbox {SLE}_\kappa $$ loop measures, we construct the unrooted $$\hbox {SLE}_\kappa $$ loop measure in $$\widehat{\mathbb {C}}$$ , which satisfies Mobius invariance and reversibility. Third, we use Brownian loop measures to extend the rooted and unrooted $$\hbox {SLE}_\kappa $$ loop measures from $$\widehat{\mathbb {C}}$$ to subdomains of $$\widehat{\mathbb {C}}$$ , which respectively satisfy conformal covariance and conformal invariance, and then further use the conformal invariance to extend unrooted $$\hbox {SLE}_\kappa $$ loop measures to some Riemann surfaces. Finally, using a similar approach, we construct $$\hbox {SLE}_\kappa $$ bubble measures in simply/multiply connected domains rooted at a boundary point. The space-time homogeneity of rooted $$\hbox {SLE}_\kappa $$ loop measures in $$\widehat{\mathbb {C}}$$ confirms a conjecture by Greg Lawler on the existence of such measures.