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The Parafermionic Observable in SLE

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Abstract

The parafermionic observable has recently been used by number of authors to study discrete models, believed to be conformally invariant and to prove convergence results for these processes to SLE (Beffara and Duminil-Copin in arXiv:1010.0526v2, 2011; Duminil-Copin and Smirnov in arXiv:1007.0575v2, 2011; Hongler and Smirnov in arXiv:1008.2645v3, 2011; Ikhlef and Cardy in J. Phys. A 42:102001, 2009; Lawler in preprint, 2011; Rajabpour and Cardy in J. Phys. A 40:14703, 2007; Riva and Cardy in J. Stat. Mech. Theory Exp., 2006; Smirnov in International Congress of Mathematicians, vol. II, pp. 1421–1451, 2006; Smirnov in Ann. Math. 172(2):1435–1467, 2010; Smirnov in Proceedings of the International Congress of Mathematicians, Hyderabad 2010, vol. I, pp. 595–621, 2010). We provide a definition for a one parameter family of continuum versions of the parafermionic observable for SLE, which takes the form of a normalized limit of expressions identical to the discrete definition. We then show the limit defining the observable exists, compute the value of the observable up to a finite multiplicative constant, and prove this constant is non-zero for a wide range of κ. Finally, we show our observable for SLE becomes a holomorphic function for a particular choice of the parameter, which provides a new point of view on a fundamental property of the discrete observable.

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Notes

  1. These values were computed in Mathematica 7 with a working precision of 20 digits.

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Acknowledgements

The author would like to thank Greg Lawler for helpful discussions during the preparation of this paper, and the anonymous reviewer for helping clarify the presentation of several points.

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Correspondence to Brent Morehouse Werness.

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Werness, B.M. The Parafermionic Observable in SLE. J Stat Phys 149, 1112–1135 (2012). https://doi.org/10.1007/s10955-012-0657-9

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