Continuum Methods and Tightness Results for Non-Simple SLE and CLE: Existence, Interactions, and Metric Approximations

This thesis presents results in the theory of Schramm-Loewner Evolution (SLE) and Conformal Loop Ensembles (CLE), which are random fractal structures describing scaling limits of interfaces in two-dimensional statistical physics models. In particular, we deal with the construction and interaction of SLEκ curves and CLEκ loops in the non-simple regime κ∈(4,8], where the curves are self-touching and CLE loops intersect themselves and each other. The first part of the thesis provides a purely continuous proof of the existence of the chordal SLE8 curve. While the existence of SLEκ for κ different than 8 as a continuous curve was established by analytical estimates on the Loewner chain, the proof in the case κ= 8 relies on discrete methods, namely the convergence of the uniform spanning tree (UST) to SLE8. We bypass discrete models by using the imaginary geometry framework, in which SLE curves arise as flow lines of the Gaussian Free Field (GFF). We work with whole-plane space-filling SLEκ parameterized by Lebesgue measure, and establish uniform estimates on its behavior as κ↑8. A tightness argument allows us to take a limit in law, yielding a continuous version of SLE8. The second part of the thesis addresses the construction of multiple SLEκ processes from partially explored CLEκ configurations for κ∈(4,8). While CLEs describe the scaling limit of the global collection of loops from discrete models with uniform boundary conditions, multiple SLE arise as scaling limits when one considers alternating boundary conditions for the discrete models and then jointly explores the corresponding interfaces. It is possible to construct them directly in the continuum setting by starting out with a CLE and achieving alternating boundary conditions by partially exploring boundary touching loops. We define multichordal CLEκ as the conditional law of the remainder of a CLEκ after partial exploration. These configurations are shown to consist of curves following random link patterns, and their law conditioned on the pattern coincides with that of multiple SLEκ. We establish existence, uniqueness, and conformal invariance of these measures and describe a local resampling mechanism in which the explored strands can be relinked in any topologically compatible way with positive probability. The final part of the thesis investigates the metric geometry of CLEκ gasket – the set of points not surrounded by any loop – in the non-simple regime κ∈(4,8). We initiate the construction of conformal covariant metrics on the gasket through a class of approximation schemes, which includes approximations to the intrinsic (chemical) and resistance distance. Our main result establishes the tightness of these approximations under a general set of axioms, and that any subsequential limit defines a non-degenerate metric on the gasket which also satisfies a list of natural properties. We conjecture that these metrics describe the scaling limit of the chemical distance resp. effective resistance metric associated with discrete models that converge in the limit to CLEκ.

Name dc:type.qualificationname

Doctor of Philosophy (PhD)

Level dc:type.qualificationlevel

Doctoral

Grantor dc:publisher.institution

University of Cambridge

Year dc:date.issued

2025

Author dc:creator

• Ambrosio, Valeria

Advisor dc:contributor.advisor

• Miller, Jason

• Probability theory
• Complex analysis
• Schramm–Loewner Evolution (SLE)
• Conformal Loop Ensemble (CLE)
• Tightness
• Metric approximations
• Random geometry

Licence

• https://www.repository.cam.ac.uk/bitstreams/747c72ed-e20e-4f8f-8223-c18b4a316ae0/download
• http://purl.org/NET/rdflicense/allrightsreserved

Language dc:language

eng

DOI dc:identifier.doi

https://doi.org/10.17863/CAM.122372

OAI identifier oai:identifier

oai:www.repository.cam.ac.uk:1810/391131