SLE and the free field: Partition functions and couplings. (English) Zbl 1204.60079
The author studies some relations between random curves in planar simply connected domains (Schramm-Loewner Evolutions) and the massless (Euclidean) free field in such a domain. identities of partition functions between different versions of Schramm-Loewner Evolutions and the free field with appropriate boundary conditions are established. This involves \(\zeta\)-regularization and the Polyakov-Alvarez conformal anomaly formula. The author proceeds with a construction of couplings of Schramm-Loewner Evolutions with the free field, showing that, in a precise sense, chordal Schramm-Loewner Evolutions is the solution of a stochastic “differential” equation driven by the free field.
Reviewer: Anatoliy Pogorui (Zhytomyr)
MSC:
| 60J67 | Stochastic (Schramm-)Loewner evolution (SLE) |
| 60G17 | Sample path properties |
| 60K35 | Interacting random processes; statistical mechanics type models; percolation theory |
References:
| [1] | Orlando Alvarez, Theory of strings with boundaries: fluctuations, topology and quantum geometry, Nuclear Phys. B 216 (1983), no. 1, 125 – 184. · doi:10.1016/0550-3213(83)90490-X |
| [2] | Michel Bauer and Denis Bernard, 2D growth processes: SLE and Loewner chains, Phys. Rep. 432 (2006), no. 3-4, 115 – 221. · doi:10.1016/j.physrep.2006.06.002 |
| [3] | Vincent Beffara, Hausdorff dimensions for \?\?\?\(_{6}\), Ann. Probab. 32 (2004), no. 3B, 2606 – 2629. · Zbl 1055.60036 · doi:10.1214/009117904000000072 |
| [4] | Federico Camia and Charles M. Newman, Two-dimensional critical percolation: the full scaling limit, Comm. Math. Phys. 268 (2006), no. 1, 1 – 38. · Zbl 1117.60086 · doi:10.1007/s00220-006-0086-1 |
| [5] | Giuseppe Da Prato and Jerzy Zabczyk, Second order partial differential equations in Hilbert spaces, London Mathematical Society Lecture Note Series, vol. 293, Cambridge University Press, Cambridge, 2002. · Zbl 1012.35001 |
| [6] | Julien Dubédat, \?\?\?(\?,\?) martingales and duality, Ann. Probab. 33 (2005), no. 1, 223 – 243. · Zbl 1096.60037 · doi:10.1214/009117904000000793 |
| [7] | Julien Dubédat, Euler integrals for commuting SLEs, J. Stat. Phys. 123 (2006), no. 6, 1183 – 1218. · Zbl 1113.82064 · doi:10.1007/s10955-006-9132-9 |
| [8] | Julien Dubédat, Excursion decompositions for SLE and Watts’ crossing formula, Probab. Theory Related Fields 134 (2006), no. 3, 453 – 488. · Zbl 1112.60032 · doi:10.1007/s00440-005-0446-3 |
| [9] | Julien Dubédat, Commutation relations for Schramm-Loewner evolutions, Comm. Pure Appl. Math. 60 (2007), no. 12, 1792 – 1847. · Zbl 1137.82009 · doi:10.1002/cpa.20191 |
| [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 (2004), no. 3, 279 – 302. · Zbl 1149.81352 · doi:10.1016/j.nuclphysb.2004.03.025 |
| [13] | Krzysztof GawÈ©dzki, Lectures on conformal field theory, Quantum fields and strings: a course for mathematicians, Vol. 1, 2 (Princeton, NJ, 1996/1997) Amer. Math. Soc., Providence, RI, 1999, pp. 727 – 805. · Zbl 1170.81430 |
| [14] | James Glimm and Arthur Jaffe, Quantum physics, 2nd ed., Springer-Verlag, New York, 1987. A functional integral point of view. · Zbl 0461.46051 |
| [15] | Svante Janson, Gaussian Hilbert spaces, Cambridge Tracts in Mathematics, vol. 129, Cambridge University Press, Cambridge, 1997. · Zbl 0887.60009 |
| [16] | Richard Kenyon, Conformal invariance of domino tiling, Ann. Probab. 28 (2000), no. 2, 759 – 795. · Zbl 1043.52014 · doi:10.1214/aop/1019160260 |
| [17] | Richard Kenyon, Dominos and the Gaussian free field, Ann. Probab. 29 (2001), no. 3, 1128 – 1137. · Zbl 1034.82021 · doi:10.1214/aop/1015345599 |
| [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] | Richard W. Kenyon, James G. Propp, and David B. Wilson, Trees and matchings, Electron. J. Combin. 7 (2000), Research Paper 25, 34. · Zbl 0939.05066 |
| [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 (2007), no. Anal. i Osob. Ch. 1, 107 – 153; English transl., Proc. Steklov Inst. Math. 258 (2007), no. 1, 100 – 146. · Zbl 1155.81367 · doi:10.1134/S0081543807030108 |
| [22] | Gregory Lawler, Oded Schramm, and Wendelin Werner, Conformal restriction: the chordal case, J. Amer. Math. Soc. 16 (2003), no. 4, 917 – 955. · Zbl 1030.60096 |
| [23] | Gregory F. Lawler, Conformally invariant processes in the plane, Mathematical Surveys and Monographs, vol. 114, American Mathematical Society, Providence, RI, 2005. · Zbl 1074.60002 |
| [24] | Gregory F. Lawler, Oded Schramm, and Wendelin Werner, Conformal invariance of planar loop-erased random walks and uniform spanning trees, Ann. Probab. 32 (2004), no. 1B, 939 – 995. · Zbl 1126.82011 · doi:10.1214/aop/1079021469 |
| [25] | Gregory F. Lawler and José A. Trujillo Ferreras, Random walk loop soup, Trans. Amer. Math. Soc. 359 (2007), no. 2, 767 – 787. · Zbl 1120.60037 |
| [26] | Gregory F. Lawler and Wendelin Werner, The Brownian loop soup, Probab. Theory Related Fields 128 (2004), no. 4, 565 – 588. · doi:10.1007/s00440-003-0319-6 |
| [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 (1988), no. 1, 148 – 211. · Zbl 0653.53022 · doi:10.1016/0022-1236(88)90070-5 |
| [29] | A. M. Polyakov, Quantum geometry of bosonic strings, Phys. Lett. B 103 (1981), no. 3, 207 – 210. , https://doi.org/10.1016/0370-2693(81)90743-7 A. M. Polyakov, Quantum geometry of fermionic strings, Phys. Lett. B 103 (1981), no. 3, 211 – 213. · doi:10.1016/0370-2693(81)90744-9 |
| [30] | Daniel Revuz and Marc Yor, Continuous martingales and Brownian motion, 3rd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 293, Springer-Verlag, Berlin, 1999. · Zbl 0917.60006 |
| [31] | Steffen Rohde and Oded Schramm, Basic properties of SLE, Ann. of Math. (2) 161 (2005), no. 2, 883 – 924. · Zbl 1081.60069 · doi:10.4007/annals.2005.161.883 |
| [32] | Oded Schramm, Scaling limits of loop-erased random walks and uniform spanning trees, Israel J. Math. 118 (2000), 221 – 288. · Zbl 0968.60093 · doi:10.1007/BF02803524 |
| [33] | O. Schramm and S. Sheffield. Contour lines of the two-dimensional discrete Gaussian free field. Acta Math., 202(1):21-137, 2009. · Zbl 1210.60051 |
| [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] | Scott Sheffield, Gaussian free fields for mathematicians, Probab. Theory Related Fields 139 (2007), no. 3-4, 521 – 541. · Zbl 1132.60072 · doi:10.1007/s00440-006-0050-1 |
| [37] | Barry Simon, The \?(\?)\(_{2}\) Euclidean (quantum) field theory, Princeton University Press, Princeton, N.J., 1974. Princeton Series in Physics. · Zbl 1175.81146 |
| [38] | Barry Simon, Trace ideals and their applications, 2nd ed., Mathematical Surveys and Monographs, vol. 120, American Mathematical Society, Providence, RI, 2005. · Zbl 1074.47001 |
| [39] | Hidenori Sonoda, Functional determinants on punctured Riemann surfaces and their application to string theory, Nuclear Phys. B 294 (1987), no. 1, 157 – 192. · doi:10.1016/0550-3213(87)90578-5 |
| [40] | Wendelin Werner, Random planar curves and Schramm-Loewner evolutions, Lectures on probability theory and statistics, Lecture Notes in Math., vol. 1840, Springer, Berlin, 2004, pp. 107 – 195. · Zbl 1057.60078 · doi:10.1007/978-3-540-39982-7_2 |
| [41] | Dapeng Zhan, Duality of chordal SLE, Invent. Math. 174 (2008), no. 2, 309 – 353. · Zbl 1158.60047 · doi:10.1007/s00222-008-0132-z |
| [42] | Dapeng Zhan, Reversibility of chordal SLE, Ann. Probab. 36 (2008), no. 4, 1472 – 1494. · Zbl 1157.60051 · doi:10.1214/07-AOP366 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
