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Stochastic Coupling and Duality for Interacting Schramm-Loewner Evolutions on Multiply Connected Domains

Year: 2025
Type: article
Abstract: The Schramm-Loewner Evolution (SLE) provides a canonical model for random non-self-intersecting curves arising as scaling limits of interfaces in two-dimensional statistical mechanics models at criticality. While the theory of a single SLE curve in simply connected domains is well-established, the behavior of multiple, interacting SLE curves, particularly in domains with non-trivial topology, presents significant challenges. This paper investigates systems of interacting SLEκ processes on multiply connected domains, with a focus on the canonical case of the annulus. We develop a rigorous framework for defining these interacting processes via a system of coupled stochastic differential equations governing their driving functions, where the interaction terms are derived from the principles of Conformal Field Theory (CFT). Our central result is the construction of a novel stochastic coupling that establishes a duality between a system of n interacting SLEκ curves growing from the inner boundary of an annulus and a corresponding system of n interacting SLE16/κ curves growing from the outer boundary. This duality is realized through a time-dependent conformal map whose evolution is intricately tied to the geometry of the growing curves. We prove the existence and uniqueness of the coupled process and explore its geometric consequences, including relationships between the fractal dimensions of the primal and dual clusters and the probabilities of specific topological connection events. This work extends the celebrated SLE duality to a more complex setting, providing new analytical tools for studying critical phenomena on non-simply connected geometries and reinforcing the profound connection between stochastic analysis and CFT.
DOI: 10.5281/zenodo.17387021
Language: English
Open Access status: green
OpenAlex ID: W7092366771
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