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

TL;DR

This work constructs the canonical Brownian motion on the CLEκ' gasket for κ' in (4,8) by developing a CLE-specific resistance form and proving its existence, uniqueness, and correspondence with a diffusion. The key advances are (i) defining weak CLEκ' resistance forms via tightness and locality, (ii) proving a bi-Lipschitz uniqueness that yields a strong CLEκ' resistance form with a unique scaling exponent, and (iii) establishing that the resulting diffusion is the CLEκ' Brownian motion, not conformally invariant due to the fractal gasket structure. The results set the stage for proving scaling-limit conjectures for lattice models (e.g., κ=6 for critical percolation) by showing convergence of resistance metrics and associated Markov processes. Together, these developments provide a rigorous diffusion framework on CLE gaskets, linking geometric, probabilistic, and analytic facets of conformally invariant fractals. The work also outlines a program to connect simple random walk limits on CLE-convergent models with CLEκ' Brownian motion in future studies.

Abstract

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

Figures (9)

• Figure 1.1: Illustration of the locality property of the ${\rm CLE}_{\kappa'}$ resistance form. The energy of $f$ is given by the sum of the energies of the restrictions of $f$ to the subregions (indicated with different colors), and the energy of $f$ in each subregion is locally determined by the ${\rm CLE}_{\kappa'}$. The subregions are connected at a finite number of points (indicated by red dots).
• Figure 3.1: Illustration of the locality property \ref{['it:weak_rform_cut_loop']} of weak ${\rm CLE}_{\kappa'}$ resistance forms. (A simplified sketch of) a loop $\mathcal{L}$ contained in $\overline{V}$ is shown in green.
• Figure 3.2: Suppose $V$ consists of the regions shaded in yellow and pink. If $V' \supseteq V$ contains the green loop, then the pink region becomes part of $V'_{(s(\epsilon))}$, so $V \setminus V_{(s(\epsilon))} \nsubseteq V' \setminus V'_{(s(\epsilon))}$. The dashed edges are also lost in $\mathfrak{G}_{\epsilon}^{V'}$.
• Figure 3.3: Top left: The region $U$ between the outer boundaries of $\eta'_1$ and $\eta'_2$ is shown in yellow. This region is used to define the renormalization constant ${\mathfrak m}_{\epsilon}^{}$. Top right: The additional dead ends in the region $U'$ between $\eta'_1$ and $\eta'_2$ are shown in grey. Note that the cables emanating from the vertices in $U' \setminus U$ create additional edges between the vertices in $U$. Bottom: The dead ends of size at most $s(\epsilon)$ are shown in pink. Removing them decreases the number of additional edges.
• Figure 3.4: Shown is a region $V$ and some of the loops in $\overline{V}$. The small dead ends in $V_{(s(\epsilon))}$ cut off by these loops are shown in pink. When we divide $V$ into subcomponents (one is shown in yellow), we need to remove the dead ends also from the subcomponents in order to make their graph approximations match.

Theorems & Definitions (139)

• Definition 1.1
• Theorem 1.2
• Definition 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Definition 2.1
• Theorem 2.2: amy-cle-resampling
• Proposition 2.3: amy-cle-resampling
• Lemma 2.5: amy-cle-resampling