CLEₖ Brownian Motion on Fractal Gaskets

• CLEₖ Brownian motion is a canonical symmetric diffusion process on CLE gaskets that exhibits scale invariance and conformal covariance.
• It utilizes resistance and Dirichlet forms to define effective metrics and characterize heat kernels via spectral dimensions.
• The construction links lattice model scaling limits with diffusive behaviors in random fractals, providing insights into critical statistical mechanics.

A CLE$_\kappa$ Brownian motion is a canonical symmetric diffusion process naturally associated with the gasket of a Conformal Loop Ensemble (CLE$_\kappa$), constructed and characterized in the regime %%%%2%%%%. The process is defined on the random fractal set arising as the gasket of CLE$_\kappa$—the set of points in a planar domain not surrounded by any CLE$\kappa$ loop—and exhibits scale invariance, translation invariance, and conformal covariance. The construction of CLE$_\kappa$ Brownian motion is motivated by the conjectural scaling limit of simple random walks on continuum analogues of random lattice models whose interfaces converge to CLE$_\kappa$ (for instance, critical percolation for $\kappa=6$) (Miller et al., 4 Dec 2025). The machinery underpinning this construction combines the theory of resistance forms on fractal spaces, Dirichlet forms, and the geometric structure of CLE gaskets.

1. CLE$_\kappa$ Gaskets and Geometric Structure

A conformal loop ensemble CLE$_\kappa$ is a random, locally finite, non-crossing collection of loops in a planar domain $D\subset\mathbb{C}$, described locally by SLE$_\kappa$-type curves. For $\kappa' \in (4,8)$, such loops can self-touch and mutually touch but cannot cross. The gasket $\Upsilon_\Gamma$ of a CLE$_{\kappa'}$ (with $\Gamma$ the loop collection) is defined as

$$\Upsilon_\Gamma = \{ z \in D : z \text{ does not lie on or in any loop of } \Gamma \}.$$

This gasket forms a closed random fractal set, whose Hausdorff dimension is given by

$$d_- = 2 - \frac{(8-\kappa')(3\kappa'-8)}{32\kappa'}.$$

A geodesic (chemical) metric $d_{\text{path}}$ is defined on the gasket by minimizing the Euclidean diameter over paths within the gasket connecting two points. The resulting metric space is almost surely complete and geodesic (Miller et al., 4 Dec 2025).

2. Resistance Form Framework on the CLE$_\kappa$ Gasket

A resistance form, in the sense of Kigami, is a symmetric bilinear form $(\mathcal{E},\mathcal{F})$ defined on functions on a set $F$, generating an effective resistance metric $R$ on $F$. The resistance metric mirrors the classical electrical resistance interpretation on networks. The specific construction on $\Upsilon_\Gamma$ involves:

• Additivity: The form is additive at cut-points where the domain is decomposed.
• Localization: Each resistance form on subdomains is locally determined by the CLE configuration in that region.
• Scale Covariance: Under scaling by $\lambda>0$, energy is scaled as $\lambda^{-\alpha}$, where $\alpha$ is a universal parameter depending only on $\kappa'$.
• Translation and Conformal Covariance: The form transforms naturally under translations and conformal maps via explicit exponents.

It is shown that, for each $\kappa' \in (4,8)$, there is a unique (modulo constant) family of resistance forms on all such gaskets satisfying these conditions, up to a deterministic scaling exponent $\alpha$ in $[d^{\prime\prime},d_+]$, where $d^{\prime\prime}$ and $d_+$ are the double-point and outer-boundary dimensions, respectively (Miller et al., 4 Dec 2025).

3. Construction and Properties of CLE$_\kappa$ Brownian Motion

Given the resistance form $(\mathcal{E},\mathcal{F})$ and a full-support Borel measure $\mu_\Gamma$ on $\Upsilon_\Gamma$ (conformally covariant with respect to $d_-$), the associated Dirichlet form gives rise to a Hunt process $X_t$ via classical theory. This process, called the CLE$_{\kappa'}$-Brownian motion, is characterized by:

• $\mu_\Gamma$-symmetry: The law is reversible with respect to $\mu_\Gamma$.
• Continuity of Paths: Sample paths are almost surely continuous with respect to the resistance metric.
• Scaling/Conformal Covariance: For any $\lambda>0$, the time-rescaled process $X_t^{(\lambda)} := \lambda X_{\lambda^{-2\alpha}t}$ is also a CLE$_\kappa$-Brownian motion on the scaled gasket; conformal covariance is defined analogously with explicit exponents.
• Local Determinism: The law of the motion in a subdomain depends only on the CLE geometry in that subdomain.
• Killed Processes: Stopping $X_t$ upon exiting a domain yields another CLE$_\kappa$-Brownian motion for that domain.
• Heat Kernel and Spectral Dimension: The transition kernel is jointly continuous, with on-diagonal upper bounds controlled by the spectral dimension $d_s = 2d_-/(d_+\alpha)$ (Miller et al., 4 Dec 2025).

4. Connections with Lattice Models and Scaling Limits

There is a conjecture that for statistical mechanics models (such as critical site percolation on the triangular lattice) whose continuum scaling limits are described by CLE$_\kappa$ for $\kappa' \in (4,8)$, the simple random walk on a large cluster, equipped with the effective resistance metric and the uniform measure, converges in the Gromov-Hausdorff-Prokhorov-resistance topology to the triple $(\Upsilon_\Gamma, R, \mu_\Gamma)$ and, consequently, the random walk itself converges in law to CLE$_\kappa$ Brownian motion. This connection generalizes the "ant-in-the-labyrinth" limit for percolation clusters ($\kappa'=6$) (Miller et al., 4 Dec 2025).

5. Scaling and Conformal Invariance Principles

The CLE$_\kappa$ Brownian motion enjoys robust invariance properties:

• Under Euclidean scaling, the process and the resistance form transform via predictable powers, and the process remains within the same class.
• Under conformal maps $\phi: D\to D'$, the image of the gasket, measure, and process are transformed via explicit exponents—$d_-$ for the measure and $\alpha$ for the resistance metric—ensuring that the CLE$_\kappa$ Brownian motion defined on one domain is mapped to that on any other conformally equivalent domain.

This invariance structures the process as a canonical, geometry-adapted diffusion for CLE gaskets, making it a central object for future developments relating continuum random fractals, scaling limits, and probabilistic models in planar statistical mechanics (Miller et al., 4 Dec 2025).

6. Broader Context: Related Constructions and Dimension Theory

The construction of canonical Brownian motion on random fractals, such as the CLE$_\kappa$ gasket, is part of a broader program linking random geometric structures, Dirichlet forms, and diffusions. Results such as the almost sure KPZ-type formula, as in the peanosphere framework for CLE/SLE-related models, allow for the computation of fractal and spectral dimensions by reducing problems to processes on planar Brownian motion sets (Gwynne et al., 2015). The construction of CLE$_\kappa$ loop ensembles via Brownian loop soups and their exploration and Markovian properties further illuminate the canonical nature of the associated Brownian motion.

---

Summary Table: Key Properties of CLE$_\kappa$ Brownian Motion

Property	Description	Source
Support	CLE$_\kappa$ gasket $\Upsilon_\Gamma$ in $D\subset\mathbb{C}$	(Miller et al., 4 Dec 2025)
Uniqueness	Unique (up to scaling) process satisfying local, scale, conformal axioms	(Miller et al., 4 Dec 2025)
Scaling exponent $\alpha$	Universal, determined by $\kappa'$, $d_-$, $d_+$	(Miller et al., 4 Dec 2025)
Symmetry	Reversible w.r.t. conformal $d_-$-measure $\mu_\Gamma$	(Miller et al., 4 Dec 2025)
Scaling/conformal covariance	Law preserved under scaling/time-change, conformal maps	(Miller et al., 4 Dec 2025)
Connection to lattice models	Conjectural scaling limit of cluster random walks	(Miller et al., 4 Dec 2025)

For $\kappa' \in (4,8)$, CLE$_\kappa$ Brownian motion thus represents the canonical diffusion process “on the gasket” and encodes both probabilistic and geometric properties intrinsic to the underlying conformal loop ensemble.