Existence and uniqueness of the canonical Brownian motion in non-simple conformal loop ensemble gaskets

Abstract

We construct the canonical Brownian motion on the gasket of conformal loop ensembles (CLE$_\kappa$) for $\kappa \in (4,8)$ (which is the range of parameter values in which loops of the CLE$_\kappa$ can intersect themselves, each other, and the domain boundary). More precisely, we show that there is a unique diffusion process on the CLE$_\kappa$ gasket whose law depends locally on the CLE$_\kappa$ and satisfies certain natural properties such as translation-invariance and scale-invariance (modulo time change). We characterize the diffusion process by its resistance form and show in particular that there is a unique resistance form on the CLE$_\kappa$ gasket that is locally determined by the CLE$_\kappa$ and satisfies certain natural properties such as translation-invariance and scale-covariance. We conjecture that the CLE$_\kappa$ Brownian motion describes the scaling limit of simple random walk on statistical mechanics models in two dimensions that converge to CLE$_\kappa$. In future work the results of this paper will be used to show that this is the case with $\kappa=6$ for critical percolation on the triangular lattice.