Abstract
The level lines of the Gaussian free field are known to be related to SLE4. It is shown how this relation allows to define chordal SLE4 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 SLE4 to this setting. This allows for a derivation of new passage probabilities à la Schramm that interpolate continuously from Dirichlet to Neumann conditions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abramowitz, M., Stegun, I.: Handbook of Mathematical Functions. Dover, New York (1970)
Bauer, M., Bernard, D.: SLE κ growth processes and conformal field theories. Phys. Lett. B 543, 135–138 (2002)
Bauer, M., Bernard, D.: Conformal Field Theories of Stochastic Loewner Evolutions. Commun. Math. Phys. 239, 493–521 (2003)
Bauer, M., Bernard, D.: SLE, CFT and zig-zag probabilities. In: Conformal Invariance and Random Spatial Processes. NATO Advanced Study Institute (2003)
Bauer, M., Bernard, D.: CFTs of SLEs: the radial case. Phys. Lett. B 583, 324–330 (2004)
Bauer, M., Bernard, D.: 2D growth processes: SLE and Loewner chains. Phys. Rep. 432, 115–221 (2006)
Bauer, M., Bernard, D., Houdayer, J.: Dipolar stochastic Loewner evolutions. J. Stat. Mech. 2005, P03001 (2005)
Bauer, M., Bernard, D., Kennedy, T.: Conditioning Schramm-Loewner evolutions and loop erased random walks. J. Math. Phys. 50, 043301 (2009)
Bauer, M., Bernard, D., Kytölä, K.: Multiple Schramm-Loewner evolutions and statistical mechanics martingales. J. Stat. Phys. 120, 1125–1163 (2005)
Bauer, R.O., Friedrich, R.M.: On radial stochastic Loewner evolution in multiply connected domains. J. Funct. Anal. 237, 565–588 (2006)
Bauer, R.O., Friedrich, R.M.: On Chordal and Bilateral SLE in multiply connected domains. Math. Z. 258, 241–265 (2008)
Øksendal, B.: Stochastic Differential Equations. Springer, Berlin (2007)
Bettelheim, E., Gruzberg, I.A., Ludwig, A.W.W., Wiegmann, P.: Stochastic Loewner evolution for conformal field theories with Lie group symmetries. Phys. Rev. Lett. 95, 251601 (2005)
Callan, C.G., Klebanov, I.R.: Exact c=1 boundary conformal field theories. Phys. Rev. Lett. 72, 1968–1971 (1994)
Callan, C.G., Klebanov, I.R., Ludwig, A.W.W., Maldacena, J.M.: Exact solution of a boundary conformal field theory. Nucl. Phys. B 422, 417–448 (1994)
Cardy, J.: SLE(κ,ρ) and conformal field theory. arXiv:math-ph/0412033
Cardy, J.: SLE for theoretical physicists. Ann. Phys. 318, 81–115 (2005)
Doob, J.L.: Classical Potential Theory and Its Probabilistic Counterpart. Springer, New York (1984)
Francesco, P.D., Mathieu, P., Sénéchal, D.: Conformal Field Theory. Springer, Berlin (1997)
Gaberdiel, M.R., Recknagel, A.: Conformal boundary states for free bosons and fermions. J. High Energy Phys. 11, 16 (2001)
Gaberdiel, M.R., Recknagel, A., Watts, G.M.T.: The conformal boundary states for SU(2) at level 1. Nucl. Phys. B 626, 344–362 (2002)
Hagendorf, C.: A generalization of Schramm’s formula for SLE(2). J. Stat. Mech. P02033 (2009)
Hagendorf, C.: Evolutions de Schramm-Loewner et théories conformes; deux exemples de systèmes désordonnés de basse dimension. Ph.D. thesis, Université Pierre et Marie Curie Paris VI (2009)
Komatu, Y.: Über einen Satz von Herrn Löwner. Proc. Imp. Acad. Tokyo 16, 512–514 (1940)
Komatu, Y.: On conformal slit mapping of multiply-connected domains. Proc. Jpn. Acad. 26, 26–31 (1950)
Lawler, G.F.: Conformally Invariant Processes in the Plane. Am. Math. Soc., Providence (2005)
Lawler, G.F., Schramm, O., Werner, W.: Conformal invariance of planar loop-erased random walks and uniform spanning trees. Ann. Probab. 32, 939–995 (2004)
Lawler, G.F., Schramm, O., Werner, W.: On the scaling limit of planar self-avoiding walk. In: Fractal Geometry and Applications: A Jubilee of Benoit Mandelbrot, Part 2. Proc. Sympos. Pure Math., vol. 72, pp. 339–364. Am. Math. Soc., Providence (2004)
Nehari, Z.: Conformal Mapping. Dover, New York (1982)
Olver, P.J.: Application of Lie Groups to Differential Equations. Springer, New York (1993)
Recknagel, A., Schomerus, V.: Boundary deformation theory and moduli spaces of D-branes. Nucl. Phys. B 545, 233–282 (1999)
Rohde, S., Schramm, O.: Basic properties of SLE. Ann. Math. 161, 883–924 (2005)
Schramm, O.: A percolation formula. Electron. Commun. Probab. 6, 115–120 (2001)
Schramm, O., Sheffield, S.: Contour lines of the two-dimensional discrete Gaussian free field. Acta Math. 202, 21–137 (2009)
Schramm, O., Wilson, D.B.: SLE coordinate changes. N.Y. J. Math. 11, 659–669 (2005)
Sheffield, S.: Exploration trees and conformal loop ensembles. Duke Math. J. 147, 79–129 (2009)
Smirnov, S.: Conformal invariance in random cluster models. I. Holomorphic fermions in the Ising model. Ann. Math. (2009, to appear). arXiv:0708.0039
Zhan, D.: Stochastic Loewner evolutions in doubly connected domains. Probab. Theory Relat. Fields 129, 340–380 (2004)
Zhan, D.: Some properties of annulus SLE. Electron. J. Probab. 11, 1069–1093 (2006)
Zhan, D.: On the reversal of radial SLE, I: Commutation Relations in Annuli (2009). arXiv:0904.0808
Author information
Authors and Affiliations
Corresponding author
Additional information
D. Bernard is a Member of the CNRS.
Rights and permissions
About this article
Cite this article
Hagendorf, C., Bernard, D. & Bauer, M. The Gaussian Free Field and SLE4 on Doubly Connected Domains. J Stat Phys 140, 1–26 (2010). https://doi.org/10.1007/s10955-010-9980-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10955-010-9980-1