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Slit Holomorphic Stochastic Flows and Gaussian Free Field

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Abstract

It was realized recently that the chordal, radial and dipolar Schramm–Löwner evolution (SLEs) are special cases of a general slit holomorphic stochastic flow. We characterize those slit holomorphic stochastic flows which generate level lines of the Gaussian free field. In particular, we describe the modifications of the Gaussian free field (GFF) corresponding to the chordal and dipolar SLE with drifts. Finally, we develop a version of conformal field theory based on the background charge and Dirichlet boundary condition modifications of GFF and present martingale-observables for these types of SLEs.

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Acknowledgments

The authors would like to thank Alexey Tochin for the discussions on the subject of this paper.

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Authors and Affiliations

  1. Department of Mathematics, University of Bergen, P.O. Box 7803, Bergen, 5020, Norway

    Georgy Ivanov & Alexander Vasil’ev

  2. School of Mathematics, Korea Institute of Advanced Study, 85 Hoegiro, Dongdaemun-gu, Seoul, 02455, Republic of Korea

    Nam-Gyu Kang

Authors
  1. Georgy Ivanov
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  2. Nam-Gyu Kang
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  3. Alexander Vasil’ev
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Corresponding author

Correspondence to Alexander Vasil’ev.

Additional information

Communicated by David Shoikhet.

G. Ivanov and A. Vasil’ev: have been supported by EU FP7 IRSES program STREVCOMS, Grant no. PIRSES-GA-2013-612669, by the Grant of the Norwegian Research Council #239033/F20. G. Ivanov: has also been supported by Meltzerfondet. N.-G. Kang: has also been supported by Samsung Science and Technology Foundation (SSTF-BA1401-01).

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Ivanov, G., Kang, NG. & Vasil’ev, A. Slit Holomorphic Stochastic Flows and Gaussian Free Field. Complex Anal. Oper. Theory 10, 1591–1617 (2016). https://doi.org/10.1007/s11785-016-0536-5

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  • DOI: https://doi.org/10.1007/s11785-016-0536-5

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