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SLE and the free field: Partition functions and couplings. (English) Zbl 1204.60079
The author studies some relations between random curves in planar simply connected domains (Schramm-Loewner Evolutions) and the massless (Euclidean) free field in such a domain. identities of partition functions between different versions of Schramm-Loewner Evolutions and the free field with appropriate boundary conditions are established. This involves $$\zeta$$-regularization and the Polyakov-Alvarez conformal anomaly formula. The author proceeds with a construction of couplings of Schramm-Loewner Evolutions with the free field, showing that, in a precise sense, chordal Schramm-Loewner Evolutions is the solution of a stochastic “differential” equation driven by the free field.

##### MSC:
 60J67 Stochastic (Schramm-)Loewner evolution (SLE) 60G17 Sample path properties 60K35 Interacting random processes; statistical mechanics type models; percolation theory
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