Remote Access Journal of the American Mathematical Society
Green Open Access

Journal of the American Mathematical Society

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

 
 

 

SLE and the free field: Partition functions and couplings


Author: Julien Dubédat
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
Published electronically: April 29, 2009
MathSciNet review: 2525778
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • 1. O. Alvarez.
    Theory of strings with boundaries: Fluctuations, topology and quantum geometry.
    Nuclear Phys. B, 216(1):125-184, 1983. MR 701643 (85e:81095)
  • 2. M. Bauer and D. Bernard.
    2D growth processes: SLE and Loewner chains.
    Phys. Rep., 432(3-4):115-221, 2006. MR 2251805 (2007g:82058)
  • 3. V. Beffara.
    Hausdorff dimensions for $ \rm SLE\sb 6$.
    Ann. Probab., 32(3B):2606-2629, 2004. MR 2078552 (2005k:60295)
  • 4. F. Camia and C. M. Newman.
    Two-dimensional critical percolation: The full scaling limit.
    Comm. Math. Phys., 268(1):1-38, 2006. MR 2249794 (2007m:82032)
  • 5. G. Da Prato and J. Zabczyk.
    Second order partial differential equations in Hilbert spaces, volume 293 of London Mathematical Society Lecture Note Series.
    Cambridge University Press, Cambridge, 2002. MR 1985790 (2004e:47058)
  • 6. J. Dubédat.
    $ {\rm SLE}(\kappa,\rho)$ martingales and duality.
    Ann. Probab., 33(1):223-243, 2005. MR 2118865 (2005j:60180)
  • 7. J. Dubédat.
    Euler integrals for commuting SLEs.
    Journal Statist. Phys., 123(6):1183-1218, 2006. MR 2253875 (2007g:82027)
  • 8. J. Dubédat.
    Excursion decompositions for SLE and Watts' crossing formula.
    Probab. Theory Related Fields, 134(3):453-488, 2006. MR 2226888 (2007d:60019)
  • 9. J. Dubédat.
    Commutation relations for SLE.
    Comm. Pure Applied Math., 60(12):1792-1847, 2007. MR 2358649 (2009d:60113)
  • 10. J. Dubédat.
    Duality of Schramm-Loewner Evolutions.
    To appear, Ann. Sci. Ecole Normale Supérieure; arXiv:math.PR/0711.1884, 2007.
  • 11. J. Dubédat.
    SLE partition functions, $ \zeta$-regularization and Virasoro representations.
    in preparation, 2007.
  • 12. R. Friedrich and J. Kalkkinen.
    On conformal field theory and stochastic Loewner evolution.
    Nuclear Phys. B, 687(3):279-302, 2004. MR 2059141 (2005b:81173)
  • 13. K. Gawedzki.
    Lectures on conformal field theory.
    In Quantum fields and strings: A course for mathematicians, Vols. 1, 2 (Princeton, NJ, 1996/1997), pages 727-805. Amer. Math. Soc., Providence, RI, 1999. MR 1701610 (2001f:81175)
  • 14. J. Glimm and A. Jaffe.
    Quantum physics.
    A functional integral point of view.
    Springer-Verlag, New York, second edition, 1987. MR 887102 (89k:81001)
  • 15. S. Janson.
    Gaussian Hilbert spaces, volume 129 of Cambridge Tracts in Mathematics.
    Cambridge University Press, Cambridge, 1997. MR 1474726 (99f:60082)
  • 16. R. Kenyon.
    Conformal invariance of domino tiling.
    Ann. Probab., 28(2):759-795, 2000. MR 1782431 (2002e:52022)
  • 17. R. Kenyon.
    Dominos and the Gaussian free field.
    Ann. Probab., 29(3):1128-1137, 2001. MR 1872739 (2002k:82039)
  • 18. R. Kenyon and D. Wilson.
    Boundary Partitions in Trees and Dimers.
    To appear, Trans. Amer. Math. Soc.; preprint, arXiv:math.PR/0608422, 2006.
  • 19. R. W. Kenyon, J. G. Propp, and D. B. Wilson.
    Trees and matchings.
    Electron. J. Combin., 7:Research Paper 25, 34 pp. (electronic), 2000. MR 1756162 (2001a:05123)
  • 20. M. Kontsevich.
    SLE, CFT, and phase boundaries.
    Arbeitstagung 2003, preprint, MPI 2003 (60).
  • 21. M. Kontsevich and Y. Suhov.
    On Malliavin measures, SLE, and CFT.
    Tr. Mat. Inst. Steklova, 258(Anal. i Osob. Ch. 1):107-153, 2007. MR 2400527
  • 22. G. Lawler, O. Schramm, and W. Werner.
    Conformal restriction: The chordal case.
    J. Amer. Math. Soc., 16(4):917-955 (electronic), 2003. MR 1992830 (2004g:60130)
  • 23. G. F. Lawler.
    Conformally invariant processes in the plane, volume 114 of Mathematical Surveys and Monographs.
    American Mathematical Society, Providence, RI, 2005. MR 2129588 (2006i:60003)
  • 24. G. F. Lawler, O. Schramm, and W. Werner.
    Conformal invariance of planar loop-erased random walks and uniform spanning trees.
    Ann. Probab., 32(1B):939-995, 2004. MR 2044671 (2005f:82043)
  • 25. G. F. Lawler and J. A. Trujillo Ferreras.
    Random walk loop soup.
    Trans. Amer. Math. Soc., 359(2):767-787 (electronic), 2007. MR 2255196 (2008k:60084)
  • 26. G. F. Lawler and W. Werner.
    The Brownian loop soup.
    Probab. Theory Related Fields, 128(4):565-588, 2004. MR 2045953 (2005f:60176)
  • 27. Y. Le Jan.
    Markov loops, determinants and Gaussian fields.
    arxiv:math.PR/0612112, 2006.
  • 28. B. Osgood, R. Phillips, and P. Sarnak.
    Extremals of determinants of Laplacians.
    J. Funct. Anal., 80(1):148-211, 1988. MR 960228 (90d:58159)
  • 29. A. M. Polyakov.
    Quantum geometry of bosonic strings.
    Phys. Lett. B, 103(3):207-210, 1981. MR 623209 (84h:81093a)
  • 30. D. Revuz and M. Yor.
    Continuous martingales and Brownian motion, volume 293 of Grundlehren der Mathematischen Wissenschaften.
    Springer-Verlag, Berlin, third edition, 1999. MR 1725357 (2000h:60050)
  • 31. S. Rohde and O. Schramm.
    Basic properties of SLE.
    Ann. of Math. (2), 161(2):883-924, 2005. MR 2153402 (2006f:60093)
  • 32. O. Schramm.
    Scaling limits of loop-erased random walks and uniform spanning trees.
    Israel J. Math., 118:221-288, 2000. MR 1776084 (2001m:60227)
  • 33. O. Schramm and S. Sheffield.
    Contour lines of the two-dimensional discrete Gaussian free field.
    Acta Math., 202(1):21-137, 2009.
  • 34. O. Schramm and S. Sheffield.
    In preparation.
    2007.
  • 35. S. Sheffield.
    Exploration trees and conformal loop ensembles.
    preprint, arXiv:math.PR/ 0609167, 2006.
  • 36. S. Sheffield.
    Gaussian free fields for mathematicians.
    Probab. Theory Related Fields, 139(3-4):521-541, 2007. MR 2322706 (2008d:60120)
  • 37. B. Simon.
    The $ P(\phi )\sb{2}$ Euclidean (quantum) field theory.
    Princeton University Press, Princeton, N.J., 1974.
    Princeton Series in Physics. MR 0489552 (58:8968)
  • 38. B. Simon.
    Trace ideals and their applications, volume 120 of Mathematical Surveys and Monographs.
    American Mathematical Society, Providence, RI, second edition, 2005. MR 2154153 (2006f:47086)
  • 39. H. Sonoda.
    Functional determinants on punctured Riemann surfaces and their application to string theory.
    Nuclear Phys. B, 294(1):157-192, 1987. MR 909430 (89b:81222)
  • 40. W. Werner.
    Random planar curves and Schramm-Loewner evolutions.
    In Lectures on probability theory and statistics, volume 1840 of Lecture Notes in Math., pages 107-195. Springer, Berlin, 2004. MR 2079672 (2005m:60020)
  • 41. D. Zhan.
    Duality of chordal SLE.
    Invent. Math., 174(2):309-353, 2008. MR 2439609
  • 42. D. Zhan.
    Reversibility of chordal SLE.
    Ann. Probab., 36(4):1472-1494, 2008. MR 2435856

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (2000): 60G17, 60K35

Retrieve articles in all journals with MSC (2000): 60G17, 60K35


Additional Information

Julien Dubédat
Affiliation: Department of Mathematics, Columbia University, 2990 Broadway, New York, NY 10027

DOI: https://doi.org/10.1090/S0894-0347-09-00636-5
Keywords: Schramm-Loewner Evolutions, free field
Received by editor(s): February 27, 2008
Published electronically: April 29, 2009
Additional Notes: The author was partially supported by NSF grant DMS0804314
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society