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The Gaussian free field and SLE\(_{4}\) on doubly connected domains. (English) Zbl 1193.82027
Summary: The level lines of the Gaussian free field are known to be related to SLE\(_{4}\). It is shown how this relation allows to define chordal SLE\(_{4}\) processes on doubly connected domains, describing traces that are anchored on one of the two boundary components. The precise nature of the processes depends on the conformally invariant boundary conditions imposed on the second boundary component. Extensions of Schramm’s formula to doubly connected domains are given for the standard Dirichlet and Neumann conditions and a relation to first-exit problems for Brownian bridges is established. For the free field compactified at the self-dual radius, the extended symmetry leads to a class of conformally invariant boundary conditions parametrised by elements of SU\((2)\). It is shown how to extend SLE\(_{4}\) to this setting. This allows for a derivation of new passage probabilities à la Schramm that interpolate continuously from Dirichlet to Neumann conditions.

MSC:
82C24 Interface problems; diffusion-limited aggregation in time-dependent statistical mechanics
60J67 Stochastic (Schramm-)Loewner evolution (SLE)
60J65 Brownian motion
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
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