Sle and the free field: Partition functions and couplings

Journal of the American Mathematical Society

Volumn 22, Issue 4, 2009, Pages 995-1054

  Dubédat, Julien ^a  

^a Howard Hughes Medical Institute   (United States)

Author keywords

Free field; Schramm loewner evolutions

Indexed keywords

EID: 77949491235     PISSN: 08940347     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0894-0347-09-00636-5     Document Type: Article

References (42)

1. O. Alvarez. Theory of strings with boundaries: Fluctuations, topology and quantum geometry. Nuclear Phys. B, 216(1):125-184, 1983. MR701643 (85e:81095)
2. M. Bauer and D. Bernard. 2D growth processes: SLE and Loewner chains. Phys. Rep., 432(34):115-221, 2006. MR2251805 (2007g:82058) (Pubitemid 44362141)
3. 6. Ann. Probab., 32 (3B) : 2606-2629, 2004. MR2078552 (2005k:60295) (Pubitemid 39258782)
4. F. Camia and C. M. Newman. Two-dimensional critical percolation: The full scaling limit. Comm. Math. Phys., 268(1):1-38, 2006. MR2249794 (2007m:82032) (Pubitemid 44496742)
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. MR1985790 (2004e:47058)
6. J. Dubédat. SLE (k, p) martingales and duality. Ann. Probab., 33(1):223-243, 2005. MR2118865 (2005j:60180)
7. J. Dubédat. Euler integrals for commuting SLEs. Journal Statist. Phys., 123(6):1183-1218, 2006. MR2253875 (2007g:82027) (Pubitemid 44325597)
8. J. Dubédat. Excursion decompositions for SLE and Watts' crossing formula. Probab. Theory Related Fields, 134(3):453-488, 2006. MR2226888 (2007d:60019)
9. J. Dubédat. Commutation relations for SLE. Comm. Pure Applied Math., 60(12):1792-1847, 2007. MR2358649 (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, ζ-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. MR2059141 (2005b:81173) (Pubitemid 38561922)
13. K. Gawȩdzki. 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. MR1701610 (2001f:81175)
14. J. Glimm and A. Jaffe. Quantum physics. A functional integral point of view. Springer-Verlag, New York, second edition, 1987. MR887102 (89k:81001)
15. S. Janson. Gaussian Hilbert spaces, volume 129 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 1997. MR1474726 (99f:60082)
16. R. Kenyon. Conformal invariance of domino tiling. Ann. Probab., 28(2):759-795, 2000. MR1782431 (2002e:52022)
17. R. Kenyon. Dominos and the Gaussian free field. Ann. Probab., 29(3):1128-1137, 2001. MR1872739 (2002k:82039) (Pubitemid 33574643)
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. MR1756162 (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. MR2400527
22. G. Lawler, O. Schramm, and W. Werner. Conformal restriction: The chordal case. J. Amer. Math. Soc., 16(4):917-955 (electronic), 2003. MR1992830 (2004g:60130) (Pubitemid 37180387)
23. G. F. Lawler. Conformally invariant processes in the plane, volume 114 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2005. MR2129588 (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. MR2044671 (2005f:82043) (Pubitemid 38545321)
25. G. F. Lawler and J. A. Trujillo Ferreras. Random walk loop soup. Trans. Amer. Math. Soc., 359(2):767-787 (electronic), 2007. MR2255196 (2008k:60084)
26. G. F. Lawler and W. Werner. The Brownian loop soup. Probab. Theory Related Fields, 128(4):565-588, 2004. MR2045953 (2005f:60176) (Pubitemid 38430515)
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. MR960228 (90d:58159)
29. A. M. Polyakov. Quantum geometry of bosonic strings. Phys. Lett. B, 103(3):207-210, 1981. MR623209 (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. MR1725357 (2000h:60050)
31. S. Rohde and O. Schramm. Basic properties of SLE. Ann. of Math. (2), 161(2):883-924, 2005. MR2153402 (2006f:60093) (Pubitemid 41179960)
32. O. Schramm. Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math., 118: 221-288, 2000. MR1776084 (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(34):521-541, 2007. MR2322706 (2008d:60120)
37. 2 Euclidean (quantum) field theory. Princeton University Press, Princeton, N. J., 1974. Princeton Series in Physics. MR0489552 (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. MR2154153 (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. MR909430 (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. MR2079672 (2005m:60020)
41. D. Zhan. Duality of chordal SLE. Invent. Math., 174(2):309-353, 2008. MR2439609
42. D. Zhan. Reversibility of chordal SLE. Ann. Probab., 36(4):1472-1494, 2008. MR2435856