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

TL;DR

This paper constructs and characterizes a unique Brownian motion on the gasket of conformal loop ensembles for , using resistance forms and invariance properties, with implications for scaling limits of lattice models.

Contribution

It establishes the existence and uniqueness of a natural Brownian motion on CLE gaskets, characterized by resistance forms and invariance properties.

Findings

Proves the existence of a unique resistance form on the CLE gasket.

Characterizes the Brownian motion by its local law and invariance properties.

Conjectures the Brownian motion as the scaling limit of random walks in 2D models.

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.