Commutation Relations for SLE in Multiply Connected Domains and the Coupling Technique

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This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).This project aims at studying the geometric properties of Schramm-Loewner evolution (SLE) introduced by Oded Schramm. SLE describes conformal invariant random fractal curves in plane domains, which are the scaling limits of many interesting two dimensional statistical lattice models at critical temperature, e.g., percolation, Ising model, Gaussian free field. The relationship between SLE and Conformal Field Theory (CFT) has been established and well understood.The study of SLE itself helps people to gain better understanding of those lattice models and CFT.The project extensively uses the so-called coupling technique, which has been successful in proving the reversibility conjecture for chordal SLE and Duplantier?s duality conjecture about the boundary of SLE. The technique enables one to construct a global coupling of two SLE curves if they commute with each other. The PI plans to use the coupling technique to study SLE defined in multiply connected domains, which are important because many lattice models are naturally defined there. The properties of these lattice models indicate that any reasonablely defined SLE must satisfy commutation relation, and so the coupling technique could be applied to construct a global commutation coupling. The project focuses on SLE defined in doubly connected domains with a few marked boundary points. These SLE naturally relate with various lattice models in these domains, and they could be used to prove the reversibility of radial SLE.