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 $\kappa \in (4,8)$. 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 $_\kappa$0, described locally by SLE$_\kappa$1-type curves. For $_\kappa$2, such loops can self-touch and mutually touch but cannot cross. The gasket $_\kappa$3 of a CLE$_\kappa$4 (with $_\kappa$5 the loop collection) is defined as

$_\kappa$6

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

$_\kappa$7

A geodesic (chemical) metric $_\kappa$8 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$9 Gasket

A resistance form, in the sense of Kigami, is a symmetric bilinear form $\kappa \in (4,8)$0 defined on functions on a set $\kappa \in (4,8)$1, generating an effective resistance metric $\kappa \in (4,8)$2 on $\kappa \in (4,8)$3. The resistance metric mirrors the classical electrical resistance interpretation on networks. The specific construction on $\kappa \in (4,8)$4 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 $\kappa \in (4,8)$5, energy is scaled as $\kappa \in (4,8)$6, where $\kappa \in (4,8)$7 is a universal parameter depending only on $\kappa \in (4,8)$8.
• 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)$9, there is a unique (modulo constant) family of resistance forms on all such gaskets satisfying these conditions, up to a deterministic scaling exponent $_\kappa$0 in $_\kappa$1, where $_\kappa$2 and $_\kappa$3 are the double-point and outer-boundary dimensions, respectively (Miller et al., 4 Dec 2025).

3. Construction and Properties of CLE$_\kappa$4 Brownian Motion

Given the resistance form $_\kappa$5 and a full-support Borel measure $_\kappa$6 on $_\kappa$7 (conformally covariant with respect to $_\kappa$8), the associated Dirichlet form gives rise to a Hunt process $_\kappa$9 via classical theory. This process, called the CLE$\kappa$0-Brownian motion, is characterized by:

• $\kappa$1-symmetry: The law is reversible with respect to $\kappa$2.
• Continuity of Paths: Sample paths are almost surely continuous with respect to the resistance metric.
• Scaling/Conformal Covariance: For any $\kappa$3, the time-rescaled process $\kappa$4 is also a CLE$\kappa$5-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 $\kappa$6 upon exiting a domain yields another CLE$\kappa$7-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 $\kappa$8 (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$9 for $_\kappa$0, 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 $_\kappa$1 and, consequently, the random walk itself converges in law to CLE$_\kappa$2 Brownian motion. This connection generalizes the "ant-in-the-labyrinth" limit for percolation clusters ($_\kappa$3) (Miller et al., 4 Dec 2025).

5. Scaling and Conformal Invariance Principles

The CLE$_\kappa$4 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 $_\kappa$5, the image of the gasket, measure, and process are transformed via explicit exponents—$_\kappa$6 for the measure and $_\kappa$7 for the resistance metric—ensuring that the CLE$_\kappa$8 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$9 gasket, is part of a broader program linking random geometric structures, Dirichlet forms, and diffusions. Results such as the almost sure KPZ-type formula, as in the peanosphere framework for CLE/SLE-related models, allow for the computation of fractal and spectral dimensions by reducing problems to processes on planar Brownian motion sets (Gwynne et al., 2015). The construction of CLE$_\kappa$0 loop ensembles via Brownian loop soups and their exploration and Markovian properties further illuminate the canonical nature of the associated Brownian motion.

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Summary Table: Key Properties of CLE$_\kappa$1 Brownian Motion

Property	Description	Source
Support	CLE$_\kappa$2 gasket $_\kappa$3 in $_\kappa$4	(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 $_\kappa$5	Universal, determined by $_\kappa$6, $_\kappa$7, $_\kappa$8	(Miller et al., 4 Dec 2025)
Symmetry	Reversible w.r.t. conformal $_\kappa$9-measure $\kappa=6$0	(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=6$1, CLE$\kappa=6$2 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.