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Browsing by Author "Aarnos, Mikko"

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  • Aarnos, Mikko (2023)
    A major innovation in statistical mechanics has been the introduction of conformal field theory in the mid 1980’s. The theory postulates the existence of conformally invariant scaling limits for many critical 2D lattice models, and then uses representation theory of a certain algebraic object that can be associated to these limits to derive exact solvability results. Providing mathematical foundations for the existence of these scaling limits has been a major ongoing project ever since, and lead to the introduction of Schramm-Löwner evolution (or SLE for short) in the early 2000’s. The core insight behind SLE is that if a conformally invariant random planar curve can be described by Löwner evolution and fulfills a condition known as the domain Markov property, it must be driven by a Wiener process with no drift. Furthermore, the variance of the Wiener process can be used to define a family SLE𝜅 of random curves which are simple, self-touching or space-filling depending on 𝜅 ≥ 0. This combination of flexibility and rigidity has allowed the scaling limits of various lattice models, such as the loop-erased random walk, the harmonic explorer, and the critical Ising model with a single interface, to be described by SLE. Once we move (for example) to the critical Ising model with multiple interfaces it turns out that the standard theory of SLE is inadequate. As such we would like establish the existence of multiple SLE to handle these more general situations. However, conformal invariance and the domain Markov property no longer guarantee uniqueness of the object so the situation is more complicated. This has led to two main approaches to the study of multiple SLE, known as the global and local approaches. Global methods are often simpler, but they often do not yield explicit descriptions of the curves. On the other hand, local methods are far more involved but as a result give descriptions of the laws of the curves. Both approaches have lead to distinct proofs that the laws of the driv- ing terms of the critical Ising model on a finitely-connected domain are described by multiple SLE3 . The aim of this thesis is to provide a proof of this result on a simply-connected domain that is simpler than the ones found in the literature. Our idea is to take the proof by local approach as our base, simplify it after restricting to a simply-connected domain, and bypass the hard part of dealing with a martingale observable. We do this by defining a function as a ratio of what are known as SLE3 partition functions, and use it as a Radon-Nikodym derivative with respect to chordal SLE3 to construct a new measure. A convergence theorem for fermionic observables shows that this measure is the scaling limit of the law of the driving term of the critical Ising model with multiple interfaces, and due to our knowledge of the Radon-Nikodym derivative an application of Girsanov’s theorem shows that the measure we constructed is just local multiple SLE3.