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Communications in Mathematical Physics

, Volume 366, Issue 2, pp 469–536 | Cite as

Global and Local Multiple SLEs for \({\kappa \leq 4}\) and Connection Probabilities for Level Lines of GFF

  • Eveliina Peltola
  • Hao WuEmail author
Article
  • 64 Downloads

Abstract

This article pertains to the classification of multiple Schramm–Loewner evolutions (SLE). We construct the pure partition functions of multiple \({{\rm SLE}_\kappa}\) with \({\kappa \in (0,4]}\) and relate them to certain extremal multiple SLE measures, thus verifying a conjecture from Bauer et al. (J Stat Phys 120(5–6):1125–1163, 2005) and Kytölä and Peltola (Commun Math Phys 346(1):237–292, 2016). We prove that the two approaches to construct multiple SLEs—the global, configurational construction of Kozdron and Lawler (Universality and renormalization, vol 50 of Fields institute communications. American Mathematical Society, Providence, 2007) and Lawler (J Stat Phys 134(5–6): 813-837, 2009) and the local, growth process construction of Bauer et al. (2005), Dubédat (Commun Pure Appl Math 60(12):1792–1847, 2007), Graham (J Stat Mech Theory 2007(3):P03008, 2007) and Kytölä and Peltola (2016)—agree. The pure partition functions are closely related to crossing probabilities in critical statistical mechanics models. With explicit formulas in the special case of \({\kappa = 4}\), we show that these functions give the connection probabilities for the level lines of the Gaussian free field (GFF) with alternating boundary data. We also show that certain functions, known as conformal blocks, give rise to multiple SLE4 that can be naturally coupled with the GFF with appropriate boundary data.

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Notes

Acknowledgements

We thank V. Beffara, G. Lawler, and W. Qian for helpful discussions on multiple SLEs. We thank M. Russkikh for useful discussions on (double-)dimer models and B. Duplantier, S. Flores, A. Karrila, K. Kytölä, and A. Sepulveda for interesting, useful, and stimulating discussions. Part of this work was completed during H.W.’s visit at the IHES, which we cordially thank for hospitality. Finally, we are grateful to the referee for careful comments on the manuscript.

References

  1. BB03.
    Bauer M., Bernard D.: Conformal field theories of stochastic Loewner evolutions. Commun. Math. Phys. 239(3), 493–521 (2003)ADSMathSciNetzbMATHGoogle Scholar
  2. BB04.
    Bauer M., Bernard D.: Conformal transformations and the SLE partition function martingale. Ann. Henri Poincaré 5(2), 289–326 (2004)ADSMathSciNetzbMATHGoogle Scholar
  3. BBK05.
    Bauer M., Bernard D., Kytölä K.: Multiple Schramm–Loewner evolutions and statistical mechanics martingales. J. Stat. Phys. 120(5-6), 1125–1163 (2005)ADSMathSciNetzbMATHGoogle Scholar
  4. BH16.
    Benoist, S., Hongler, C.: The scaling limit of critical Ising interfaces is CLE(3). Ann. Probab. arXiv:1604.06975, (2019) (to appear)
  5. BPW18.
    Beffara, V., Peltola, E., Wu, H.: On the uniqueness of global multiple SLEs. Preprint in arXiv:1801.07699, (2018)
  6. BPZ84a.
    Belavin A.A., Polyakov A.M., Zamolodchikov A.B.: Infinite conformal symmetry in two-dimensional quantum field theory. Nucl. Phys. B 241(2), 333–380 (1984)ADSMathSciNetzbMATHGoogle Scholar
  7. BPZ84b.
    Belavin A.A., Polyakov A.M., Zamolodchikov A.B.: Infinite conformal symmetry of critical fluctuations in two dimensions. J. Stat. Phys. 34(5-6), 763–774 (1984)ADSMathSciNetGoogle Scholar
  8. CN07.
    Camia F., Newman C.M.: Critical percolation exploration path and SLE6: a proof of convergence. Probab. Theory Relat. Fields 139(3-4), 473–519 (2007)zbMATHGoogle Scholar
  9. Car84.
    Cardy J.L.: Conformal invariance and surface critical behavior. Nucl. Phys. B 240(4), 514–532 (1984)ADSGoogle Scholar
  10. Car89.
    Cardy J.L.: Boundary conditions, fusion rules and the Verlinde formula. Nucl. Phys. B 324(3), 581–596 (1989)ADSMathSciNetGoogle Scholar
  11. CDCH+14.
    Dmitry C., Hugo D.-C., Clément H., Antti K., Stanislav S.: Convergence of Ising interfaces to Schramm’s SLE curves. C. R. Math. 352(2), 157–161 (2014)MathSciNetzbMATHGoogle Scholar
  12. CHI15.
    Chelkak D., Hongler C., Izyurov K.: Conformal invariance of spin correlations in the planar Ising model. Ann. Math. 181(3), 1087–1138 (2015)MathSciNetzbMATHGoogle Scholar
  13. CI13.
    Chelkak D., Izyurov K.: Holomorphic spinor observables in the critical Ising model. Commun. Math. Phys. 322(2), 303–332 (2013)ADSMathSciNetzbMATHGoogle Scholar
  14. CS12.
    Chelkak D., Smirnov S.: Universality in the 2D Ising model and conformal invariance of fermionic observables. Invent. Math. 189(3), 515–580 (2012)ADSMathSciNetzbMATHGoogle Scholar
  15. DFMS97.
    Di Francesco P., Mathieu P.: Sénéchal David.: Conformal Field Theory. Springer, New York (1997)Google Scholar
  16. DF85.
    Dotsenko V.S., Fateev V.A.: Four-point correlation functions and the operator algebra in 2D conformal invariant theories with \({c \geq 1}\). Nucl. Phys. B 251, 691–734 (1985)ADSGoogle Scholar
  17. Dub06.
    Dubédat J.: Euler integrals for commuting SLEs. J. Stat. Phys. 123(6), 1183–1218 (2006)ADSMathSciNetzbMATHGoogle Scholar
  18. Dub07.
    Dubédat J.: Commutation relations for Schramm–Loewner evolutions. Commun. Pure Appl. Math. 60(12), 1792–1847 (2007)MathSciNetzbMATHGoogle Scholar
  19. Dub09.
    Dubédat J.: SLE and the free field: partition functions and couplings. J. Am. Math. Soc. 22(4), 995–1054 (2009)MathSciNetzbMATHGoogle Scholar
  20. Dub15a.
    Dubédat J.: SLE and Virasoro representations: localization. Commun. Math. Phys. 336(2), 695–760 (2015)ADSMathSciNetzbMATHGoogle Scholar
  21. Dub15b.
    Dubédat J.: SLE and Virasoro representations: fusion. Commun. Math. Phys. 336(2), 761–809 (2015)ADSMathSciNetzbMATHGoogle Scholar
  22. DS87.
    Duplantier B., Saleur H.: Exact critical properties of two-dimensional dense self-avoiding walks. Nucl. Phys. B 290, 291–326 (1987)ADSMathSciNetGoogle Scholar
  23. DS11.
    Duplantier B., Sheffield S.: Liouville quantum gravity and KPZ. Invent. Math. 185(2), 333–393 (2011)ADSMathSciNetzbMATHGoogle Scholar
  24. FFK89.
    Felder G., Fröhlich J., Keller G.: Braid matrices and structure constants for minimal conformal models. Commun. Math. Phys. 124(4), 647–664 (1989)ADSMathSciNetzbMATHGoogle Scholar
  25. FL13.
    Field L.S., Lawler G.F.: Reversed radial SLE and the Brownian loop measure. J. Stat. Phys. 150, 1030–1062 (2013)ADSMathSciNetzbMATHGoogle Scholar
  26. FK15a.
    Flores S.M., Kleban P.: A solution space for a system of null-state partial differential equations: part 1. Commun. Math. Phys., 333(1):389–434 (2015)Google Scholar
  27. FK15b.
    Flores S.M., Kleban P.: A solution space for a system of null-state partial differential equations: part 2. Commun. Math. Phys. 333(1), 435–481 (2015)ADSMathSciNetzbMATHGoogle Scholar
  28. FK15c.
    Flores S.M., Kleban P.: A solution space for a system of null-state partial differential equations: part 3. Commun. Math. Phys. 333(2), 597–667 (2015)ADSMathSciNetzbMATHGoogle Scholar
  29. FK15d.
    Flores S.M., Kleban Peter A.: solution space for a system of null-state partial differential equations: part 4. Commun. Math. Phys. 333(2), 669–715 (2015)ADSMathSciNetzbMATHGoogle Scholar
  30. FP18+.
    Flores, Steven M., Peltola, Eveliina.: Monodromy invariant CFT correlation functions of first column Kac operators. In preparation.Google Scholar
  31. FSK15.
    Flores, S.M., Simmons, J.J.H., Kleban, P.: Multiple-SLE\({_\kappa}\) connectivity weights for rectangles, hexagons, and octagons. Preprint in arXiv:1505.07756, (2015)
  32. FSKZ17.
    Flores S.M., Simmons Jacob J.H., Kleban P., Ziff R.M.: A formula for crossing probabilities of critical systems inside polygons. J. Phys. A 50(6), 064005 (2017)ADSMathSciNetzbMATHGoogle Scholar
  33. Fri04.
    Roland Friedrich. On connections of conformal field theory and stochastic Loewner evolution. Preprint in arXiv:math-ph/0410029, (2004)
  34. FK04.
    Friedrich R., Kalkkinen J.: On conformal field theory and stochastic Loewner evolution. Nucl. Phys. B 687(3), 279–302 (2004)ADSMathSciNetzbMATHGoogle Scholar
  35. FW03.
    Friedrich R., Werner W.: Conformal restriction, highest weight representations and SLE. Commun. Math. Phys. 243(1), 105–122 (2003)ADSMathSciNetzbMATHGoogle Scholar
  36. Gra07.
    Graham, K.: On multiple Schramm–Loewner evolutions. J. Stat. Mech. Theory Exp. 2007(3):P03008 (2007)Google Scholar
  37. HK13.
    Hongler C., Kytölä K.: Ising interfaces and free boundary conditions. J. Am. Math. Soc. 26(4), 1107–1189 (2013)MathSciNetzbMATHGoogle Scholar
  38. HS13.
    Hongler C., Smirnov S.: The energy density in the planar Ising model. Acta Math. 211(2), 191–225 (2013)MathSciNetzbMATHGoogle Scholar
  39. Hor67.
    Hörmander L.: Hypoelliptic second-order differential equations. Acta Math. 119, 147–171 (1967)MathSciNetzbMATHGoogle Scholar
  40. Hor90.
    Hörmander, L.: The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis, Volume 256 of Grundlehren der mathematischen Wissenschaften, 2nd edn. Springer, Berlin (1990)Google Scholar
  41. Izy15.
    Izyurov K.: Smirnov’s observable for free boundary conditions, interfaces and crossing probabilities. Commun. Math. Phys. 337(1), 225–252 (2015)ADSMathSciNetzbMATHGoogle Scholar
  42. Izy17.
    Izyurov K.: Critical Ising interfaces in multiply-connected domains. Probab. Theory Relat. Fields 167(1), 379–415 (2017)MathSciNetzbMATHGoogle Scholar
  43. JL18.
    Jahangoshahi M., Lawler Gregory F.: On the smoothness of the partition function for multiple Schramm–Loewner evolutions. J. Stat. Phys., 173(5):1353–1368 (2018)Google Scholar
  44. Ken08.
    Kenyon R.W.: Height fluctuations in the honeycomb dimer model. Commun. Math. Phys. 281(3), 675–709 (2008)ADSMathSciNetzbMATHGoogle Scholar
  45. KKP17a.
    Karrila, A., Kytölä, K., Peltola, E.: Boundary correlations in planar LERW and UST. Preprint in arXiv:1702.03261, (2017)
  46. KKP17b.
    Karrila, Alex, Kytölä, Kalle.: and Eveliina Peltola. Conformal blocks, q-combinatorics, and quantum group symmetry. Ann. Henri Poincaré D. arXiv:1709.00249, (2019) (to appear)
  47. KL05.
    Kozdron M.J., Lawler G.F.: Estimates of random walk exit probabilities and application to loop-erased random walk. Electron. J. Probab. 10(44), 1442–1467 (2005)MathSciNetzbMATHGoogle Scholar
  48. KL07.
    Kozdron, M.J., Lawler, G.F.: The configurational measure on mutually avoiding SLE paths. In: Universality and Renormalization, Volume 50 of Fields Institute Communications, pp. 199–224. American Mathematical Society, Providence (2007)Google Scholar
  49. Kon03.
    Kontsevich M.: CFT, SLE, and phase boundaries. Oberwolfach Arbeitstagung, (2003)Google Scholar
  50. KP16.
    Kytölä K., Peltola E.: Pure partition functions of multiple SLEs. Commun. Math. Phys. 346(1), 237–292 (2016)ADSMathSciNetzbMATHGoogle Scholar
  51. KP18.
    Kytölä, Kalle, Peltola, Eveliina.: Conformally covariant boundary correlation functions with a quantum group. To appear in J. Eur. Math. Soc., (2018). Preprint in arXiv:1408.1384
  52. KS18.
    Kemppainen A., Smirnov S.: Configurations of FK Ising interfaces and hypergeometric SLE. Math. Res. Lett. 25(3), 875–889 (2018)MathSciNetzbMATHGoogle Scholar
  53. KW11a.
    Kenyon R.W., Wilson D.B.: Boundary partitions in trees and dimers. Trans. Am. Math. Soc. 363(3), 1325–1364 (2011)MathSciNetzbMATHGoogle Scholar
  54. KW11b.
    Kenyon R.W., Wilson D.B.: Double-dimer pairings and skew Young diagrams. Electron. J. Combin. 18(1), 130–142 (2011)MathSciNetzbMATHGoogle Scholar
  55. Kyt07.
    Kytölä K.: Virasoro module structure of local martingales of SLE variants. Rev. Math. Phys. 19(5), 455–509 (2007)MathSciNetzbMATHGoogle Scholar
  56. Law05.
    Lawler, G.F.: Conformally Invariant Processes in the Plane, Volume 114 of Mathematical Surveys and Monographs. American Mathematical Society, Providence (2005)Google Scholar
  57. Law09a.
    Lawler G.F.: Partition functions, loop measure, and versions of SLE. J. Stat. Phys. 134(5-6), 813–837 (2009)ADSMathSciNetzbMATHGoogle Scholar
  58. Law09b.
    Lawler, Gregory F.: Schramm–Loewner evolution (SLE). In: Sheffield, S., Spencer, T. (eds.) Statistical Mechanics, Volume 16 of AMS IAS/Park City Mathematics Series (2009)Google Scholar
  59. Law11.
    Lawler, G.F.: Defining SLE in multiply connected domains with the Brownian loop measure. Preprint in arXiv:1108.4364, (2011)
  60. LSW03.
    Lawler G.F., Schramm O., Werner W.: Conformal restriction: the chordal case. J. Am. Math. Soc. 16(4), 917– (2003)Google Scholar
  61. LSW04.
    Lawler G.F., Schramm O., Werner W.: Conformal invariance of planar loop-erased random walks and uniform spanning trees. Ann. Probab. 32(1B), 939–995 (2004)MathSciNetzbMATHGoogle Scholar
  62. LW04.
    Lawler G.F., Werner W.: The Brownian loop soup. Probab. Theory Relat. Fields 128(4), 565–588 (2004)MathSciNetzbMATHGoogle Scholar
  63. LV17.
    Lenells Jonatan, Viklund Fredrik Coulomb gas integrals for commuting SLEs: Schramm’s formula and Green’s function. Preprint in arXiv:1701.03698, (2017)
  64. MS16.
    Miller J., Sheffield S.: Imaginary geometry I: interacting SLEs. Probab. Theory Relat. Fields 164(3-4), 553–705 (2016)MathSciNetzbMATHGoogle Scholar
  65. Nie87.
    Nienhuis, B.: Coulomb gas formulation of two-dimensional phase transitions. In Domb, C., Lebowitz, J.L. (eds.) Volume 11 of Phase Transitions and Critical Phenomena, pp. 1–53. Academic Press, London (1987)Google Scholar
  66. PW18.
    Peltola, E., Wu, H.: Crossing probabilities of multiple Ising interfaces. Preprint in arXiv:1808.09438 (2018)
  67. Rib14.
    Ribault, S.: Conformal field theory on the plane. Preprint in arXiv:1406.4290 (2014)
  68. RS05.
    Rohde S., Schramm O.: Basic properties of SLE. Ann. Math. 161(2), 883–924 (2005)MathSciNetzbMATHGoogle Scholar
  69. RY99.
    Revuz, D., Yor, M.: Continuous martingales and Brownian motion, volume 293 of Grundlehren der mathematischen Wissenschaften. Springer, Berlin Heidelberg, 3rd edn (1999)Google Scholar
  70. Rud91.
    Rudin, W.: Functional Analysis. International Series in Pure and Applied Mathematics, 2nd edn. McGeaw-Hill (1991)Google Scholar
  71. Sch00.
    Schramm O.: Scaling limits of loop-erased random walks and uniform spanning trees. Isr. J. Math. 118, 221–288 (2000)MathSciNetzbMATHGoogle Scholar
  72. SS09.
    Schramm O., Sheffield S.: Contour lines of the two-dimensional discrete Gaussian free field. Acta Math. 202(1), 21–137 (2009)MathSciNetzbMATHGoogle Scholar
  73. SS13.
    Schramm O., Sheffield S.: A contour line of the continuum Gaussian free field. Probab. Theory Relat. Fields 157(1-2), 47–80 (2013)MathSciNetzbMATHGoogle Scholar
  74. SW05.
    Schramm, O., Wilson, D.B. (2005) SLE coordinate changes. New York J. Math. 11, 659–669 (electronic)Google Scholar
  75. SZ12.
    Shigechi K., Zinn-Justin P.: Path representation of maximal parabolic Kazhdan–Lusztig polynomials. J. Pure Appl. Algebra 216(11), 2533–2548 (2012)MathSciNetzbMATHGoogle Scholar
  76. She07.
    Sheffield S.: Gaussian free fields for mathematicians. Probab. Theory Relat. Fields 139(3-4), 521–541 (2007)MathSciNetzbMATHGoogle Scholar
  77. Smi01.
    Smirnov S.: Critical percolation in the plane: conformal invariance, Cardy’s formula, scaling limits. C. R. Acad. Sci. Paris Sér. I Math. 333(3), 239–244 (2001)ADSMathSciNetzbMATHGoogle Scholar
  78. Smi06.
    Smirnov, S.: Towards conformal invariance of 2D lattice models. In International Congress of Mathematicians. Vol. II, pp. 1421–1451. Eur. Math. Soc. Zürich (2006)Google Scholar
  79. Smi10.
    Smirnov S.: Conformal invariance in random cluster models. I. Holomorphic fermions in the Ising model. Ann. Math. 172(2), 1435–1467 (2010)MathSciNetzbMATHGoogle Scholar
  80. Str08.
    Stroock, D.W.: An introduction to partial differential equations for probabilists, volume 112 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (2008)Google Scholar
  81. Tao09.
    Tao, T.: An epsilon of room, I: Real analysis. Pages from year three of a mathematical blog, volume 117 of Graduate Studies in Mathematics. American Mathematical Society, Providence (2009)Google Scholar
  82. Wer04.
    Werner W.: Girsanov’s transformation for \({{\rm SLE}(\kappa,\rho)}\) processes, intersection exponents and hiding exponents. Ann. Fac. Sci. Toulouse Math. (6) 13(1), 121–147 (2004)MathSciNetGoogle Scholar
  83. Wu17.
    Wu, H.: Hypergeometric SLE: conformal Markov characterization and applications. Preprint in arXiv:1703.02022 (2017)
  84. WW17.
    Wang M., Wu H.: Level lines of Gaussian free field I: zero-boundary GFF. Stoch. Process. Appl. 127(4), 1045–1124 (2017)MathSciNetzbMATHGoogle Scholar
  85. Wu18.
    Wu H.: Alternating arm exponents for the critical planar Ising model. Ann. Probab. 46(5), 2863–2907 (2018)MathSciNetzbMATHGoogle Scholar
  86. Zha08a.
    Zhan D.: Reversibility of chordal SLE. Ann. Probab. 36(4), 1472–1494 (2008)MathSciNetzbMATHGoogle Scholar
  87. Zha08b.
    Zhan D.: The scaling limits of planar LERW in finitely connected domains. Ann. Probab. 36(2), 467–529 (2008)MathSciNetzbMATHGoogle Scholar

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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Section de MathématiquesUniversité de GenèveGenevaSwitzerland
  2. 2.Yau Mathematical Sciences CenterTsinghua UniversityBeijingChina

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