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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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SLE and the free field: Partition functions and couplings
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by Julien Dubédat PDF
J. Amer. Math. Soc. 22 (2009), 995-1054 Request permission

Abstract:

Schramm-Loewner Evolutions (SLE) are random curves in planar simply connected domains; the massless (Euclidean) free field in such a domain is a random distribution. Both have conformal invariance properties in law. In the present article, some relations between the two objects are studied. We establish identities of partition functions between different versions of SLE and the free field with appropriate boundary conditions; this involves $\zeta$-regularization and the Polyakov-Alvarez conformal anomaly formula. We proceed with a construction of couplings of SLE with the free field, showing that, in a precise sense, chordal SLE is the solution of a stochastic “differential” equation driven by the free field. Existence, uniqueness in law, and pathwise uniqueness for these SDEs are proved for general $\kappa >0$. This identifies SLE curves as local observables of the free field.
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Additional Information
  • Julien Dubédat
  • Affiliation: Department of Mathematics, Columbia University, 2990 Broadway, New York, NY 10027
  • MR Author ID: 710651
  • Received by editor(s): February 27, 2008
  • Published electronically: April 29, 2009
  • Additional Notes: The author was partially supported by NSF grant DMS0804314
  • © Copyright 2009 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: J. Amer. Math. Soc. 22 (2009), 995-1054
  • MSC (2000): Primary 60G17, 60K35
  • DOI: https://doi.org/10.1090/S0894-0347-09-00636-5
  • MathSciNet review: 2525778