2025/12/04 by Miller, Jason, Yuan, Yizheng
Mathematics · Physics and Astronomy · #Stochastic processes and statistical mechanics #Random Matrices and Applications #Theoretical and Computational Physics
paper · doi:10.48550/arxiv.2512.04807
We construct the canonical Brownian motion on the gasket of conformal loop ensembles (CLEκ) for κ∈ (4,8) (which is the range of parameter values in which loops of the CLEκ can intersect themselves, each other, and the domain boundary). More precisely, we show that there is a unique diffusion process on the CLEκ gasket whose law depends locally on the CLEκ 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 CLEκ gasket that is locally determined by the CLEκ and satisfies certain natural properties such as translation-invariance and scale-covariance. We conjecture that the CLEκ Brownian motion describes the scaling limit of simple random walk on statistical mechanics models in two dimensions that converge to CLEκ. 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.