Journal of Statistical Physics

, Volume 115, Issue 5–6, pp 1149–1229 | Cite as

A Guide to Stochastic Löwner Evolution and Its Applications

  • Wouter Kager
  • Bernard Nienhuis


This article is meant to serve as a guide to recent developments in the study of the scaling limit of critical models. These new developments were made possible through the definition of the Stochastic Löwner Evolution (SLE) by Oded Schramm. This article opens with a discussion of Löwner's method, explaining how this method can be used to describe families of random curves. Then we define SLE and discuss some of its properties. We also explain how the connection can be made between SLE and the discrete models whose scaling limits it describes, or is believed to describe. Finally, we have included a discussion of results that were obtained from SLE computations. Some explicit proofs are presented as typical examples of such computations. To understand SLE sufficient knowledge of conformal mapping theory and stochastic calculus is required. This material is covered in the appendices.

scaling limits critical exponents conformal invariance conformal mappings stochastic processes Löwner's equation 


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  1. 1.
    L. V. Ahlfors, Complex Analysis: An Introduction to the Theory of Analytic Functions of One Complex Variable, 2nd ed. (McGraw–Hill, New York, 1966).Google Scholar
  2. 2.
    L. V. Ahlfors, Conformal Invariants: Topics in Geometric Function Theory (McGraw–Hill, New York, 1973).Google Scholar
  3. 3.
    R. B. Ash and C. A. Doléans-Dade, Probability & Measure Theory, 2nd ed. (Academic Press, San Diego, 2000).Google Scholar
  4. 4.
    M. Bauer and D. Bernard, SLEϰ growth processes and conformal field theories, Phys. Lett. B 543:135–138 (2002); arXiv: math-ph/0206028.Google Scholar
  5. 5.
    M. Bauer and D. Bernard, Conformal field theories of Stochastic Loewner evolutions, Comm. Math. Phys. 239:493–521 (2003); arXiv: hep-th/0210015.Google Scholar
  6. 6.
    M. Bauer and D. Bernard, SLE martingales and the Virasoro algebra, Phys. Lett. B 557:309–316 (2003); arXiv: hep-th/0301064.Google Scholar
  7. 7.
    M. Bauer and D. Bernard, Conformal transformations and the SLE partition function martingale, arXiv: math-ph/0305061.Google Scholar
  8. 8.
    R. J. Baxter, Exactly Solved Models in Statistical Mechanics (Academic Press, London, 1982).Google Scholar
  9. 9.
    R. J. Baxter, q colourings of the triangular lattice, J. Phys. A 19:2821–2839 (1986).Google Scholar
  10. 10.
    R. J. Baxter, S. B. Kelland, and F. Y. Wu, Equivalence of the Potts model or Whitney polynomial with an ice-type model, J. Phys. A 9:397–406 (1976).Google Scholar
  11. 11.
    V. Beffara, Hausdorff dimensions for SLE6 (2002); arXiv: math.PR/0204208.Google Scholar
  12. 12.
    V. Beffara, The dimension of the SLE curves (2002); arXiv: math.PR/0211322.Google Scholar
  13. 13.
    F. Camia and C. M. Newman, Continuum nonsimple loops and 2D critical percolation (2003); arXiv: math.PR/0308122.Google Scholar
  14. 14.
    J. Cardy, Conformal invariance, in Phase Transitions and Critical Phenomena, Vol. 11, C. Domb and J. L. Lebowitz, eds. (Academic Press, London, 1987), pp. 55–126.Google Scholar
  15. 15.
    J. Cardy, Critical percolation in finite geometries, J. Phys. A 25:L201-L206 (1992).Google Scholar
  16. 16.
    J. Cardy, Stochastic Loewner evolution and Dyson's circular ensembles, J. Phys. A 36:L379-L408 (2003); arXiv: math-ph/0301039.Google Scholar
  17. 17.
    B. Duplantier, Harmonic measure exponents for two-dimensional percolation, Phys. Rev. L. 82:3940–3943 (1999).Google Scholar
  18. 18.
    B. Duplantier, Conformally invariant fractals and potential theory, Phys. Rev. L. 84:1363–1367 (2000).Google Scholar
  19. 19.
    B. Duplantier and K-H. Kwon, Conformal invariance and intersections of random walks, Phys. Rev. L. 61:2514–2517 (1988).Google Scholar
  20. 20.
    R. Friedrich and W. Werner, Conformal fields, restriction properties, degenerate representations and SLE, C. R. Acad. Sci. Paris Sér. I 335:947–952 (2002); arXiv: math.PR/0209382.Google Scholar
  21. 21.
    R. Friedrich and W. Werner, Conformal restriction, highest-weight representations and SLE, Comm. Math. Phys. (2003), to appear; arXiv: math-ph/0301018.Google Scholar
  22. 22.
    C. M. Fortuin and P. W. Kasteleyn, On the random cluster model 1: Introduction and relation to other models, Physica 57:536–564 (1972).Google Scholar
  23. 23.
    T. W. Gamelin, Complex Analysis (Springer-Verlag, New York, 2000).Google Scholar
  24. 24.
    C. W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry, and the Natural Sciences (Springer-Verlag, New York, 1983).Google Scholar
  25. 25.
    G. Grimmett and D. Stirzaker, Probability and Random Processes, 3rd ed. (Oxford University Press, Oxford, 2001).Google Scholar
  26. 26.
    L. P. Kadanoff, Scaling laws for Ising models near Tc, Physics 2:263–271 (1966).Google Scholar
  27. 27.
    T. Kennedy, Monte Carlo tests of SLE predictions for the 2D self-avoiding walk (2001); arXiv: math.PR/0112246.Google Scholar
  28. 28.
    T. Kennedy, Conformal invariance and Stochastic Loewner evolution predictions for the 2D self-avoiding walk—Monte Carlo tests (2002); arXiv: math.PR/0207231.Google Scholar
  29. 29.
    G. F. Lawler, Introduction to Stochastic Processes (Chapman & Hall, New York, 1995).Google Scholar
  30. 30.
    G. F. Lawler, Hausdorff dimension of cut points for Brownian motion, Electron. J. Probab. 1:1–20 (1996).Google Scholar
  31. 31.
    G. F. Lawler, The dimension of the frontier of planar Brownian motion, Electron. Comm. Probab. 1:29–47 (1996).Google Scholar
  32. 32.
    G. F. Lawler, Geometric and fractal properties of Brownian motion and random walk paths in two and three dimensions, in Random Walks (Budapest, 1998), Bolyai Society Mathematical Studies, Vol. 9 (1999), pp. 219–258.Google Scholar
  33. 33.
    G. F. Lawler, An introduction to the Stochastic Loewner evolution; available online at URL∼jose/papers.html (2001).Google Scholar
  34. 34.
    G. F. Lawler and W. Werner, Intersection exponents for planar Brownian motion, Ann. Probab. 27:1601–1642 (1999).Google Scholar
  35. 35.
    G. F. Lawler, O. Schramm, and W. Werner, Values of Brownian intersection exponents I: Half-plane exponents, Acta Math. 187:237–273 (2001); arXiv: math.PR/9911084.Google Scholar
  36. 36.
    G. F. Lawler, O. Schramm, and W. Werner, Values of Brownian intersection exponents II: Plane exponents, Acta Math. 187:275–308 (2001); arXiv: math.PR/0003156.Google Scholar
  37. 37.
    G. F. Lawler, O. Schramm, and W. Werner, Values of Brownian intersection exponents III: Two-sided exponents, Ann. Inst. H. Poincaré Statist. 38:109–123 (2002); arXiv: math.PR/0005294.Google Scholar
  38. 38.
    G. F. Lawler, O. Schramm, and W. Werner, Analyticity of intersection exponents for planar Brownian motion, Acta Math. 189:179–201 (2002); arXiv: math.PR/0005295.Google Scholar
  39. 39.
    G. F. Lawler, O. Schramm, and W. Werner, One-arm exponent for critical 2D percolation, Electron. J. Probab. 7:13(2001); arXiv: math.PR/0108211.Google Scholar
  40. 40.
    G. F. Lawler, O. Schramm, and W. Werner, Conformal invariance of planar loop-erased random walks and uniform spanning trees, Ann. Probab. (2001), to appear; arXiv: math.PR/0112234.Google Scholar
  41. 41.
    G. F. Lawler, O. Schramm, and W. Werner, On the scaling limit of planar self-avoiding walk, in Fractal Geometry and Application. A Jubilee of Benoit Mandelbrot, AMS Proc. Symp. Pure Math. (2002), to appear; arXiv: math.PR/0204277.Google Scholar
  42. 42.
    G. F. Lawler, O. Schramm, and W. Werner, Conformal restriction: The chordal case, J. Amer. Math. Soc. 16:917–955 (2003); arXiv: math.PR/0209343.Google Scholar
  43. 43.
    K. Löwner, Untersuchungen über schlichte konforme Abbildungen des Einheitskreises. I, Math. Ann. 89:103–121 (1923).Google Scholar
  44. 44.
    S-K. Ma, Modern theory of critical phenomena, in Frontiers in Physics, Vol. 46 (Benjamin, Reading, 1976).Google Scholar
  45. 45.
    B. Nienhuis, Exact critical point and exponents of the O(n) model in two dimensions, Phys. Rev. L. 49:1062–1065 (1982).Google Scholar
  46. 46.
    B. Nienhuis, Critical behavior of two-dimensional spin models and charge asymmetry in the Coulomb Gas, J. Stat. Phys. 34:731–761 (1984). B. NienhuisCoulomb Gas formulation of two-dimensional phase transitions, in Phase Transitions and Critical Phenomena, Vol. 11, C. Domb and J. L. Lebowitz, eds. (Academic Press, London, 1987), pp. 1–53.Google Scholar
  47. 47.
    B. Nienhuis, Locus of the tricritical transition in a two-dimensional q-state Potts model, Physica A 177:109–113 (1991).Google Scholar
  48. 48.
    F. Oberhettinger, Hypergeometric functions, in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 10th ed., Chap. 15, M. Abramowitz and I. Stegun, eds. (Wiley, New York, 1972), pp. 555–566.Google Scholar
  49. 49.
    P. Pfeuty and G. Toulouse, Introduction to the Renormalization Group and to Critical Phenomena (Wiley, New York, 1977).Google Scholar
  50. 50.
    C. Pommerenke, Boundary Behaviour of Conformal Maps (Springer-Verlag, New York, 1992).Google Scholar
  51. 51.
    S. Rohde and O. Schramm, Basic properties of SLE, Ann. Math. (2001), to appear; arXiv: math.PR/0106036.Google Scholar
  52. 52.
    H. Saleur and B. Duplantier, Exact determination of the percolation hull exponent in two dimensions, Phys. Rev. Lett. 58:2325–2328 (1987).Google Scholar
  53. 53.
    O. Schramm, Scaling limits of loop-erased random walks and uniform spanning trees, Israel J. Math. 118:221–288 (2000); arXiv: math.PR/9904022.Google Scholar
  54. 54.
    O. Schramm, A percolation formula, Electron. Comm. Probab. 6:115–120 (2001); arXiv: math.PR/0107096.Google Scholar
  55. 55.
    O. Schramm and S. Sheffield, The harmonic explorer and its convergence to SLE(4) (2003); arXiv: math.PR/0310210.Google Scholar
  56. 56.
    S. Smirnov, Critical percolation in the plane: Conformal invariance, Cardy's formula, scaling limits, C. R. Acad. Sci. Paris Sér. I Math. 333:239–244 (2001). S. SmirnovA longer version is available at URL∼stas/papers//Google Scholar
  57. 57.
    S. Smirnov and W. Werner, Critical exponents for two-dimensional percolation, Math. Res. Lett. 8:729–744 (2001); arXiv: math.PR/0109120.Google Scholar
  58. 58.
    W. Werner, Random planar curves and Schramm–Löwner evolutions, Lecture Notes from the 2002 Saint-Flour Summer School (Springer, 2003), to appear; arXiv: math.PR/0303354.Google Scholar
  59. 59.
    W. Werner, Conformal restriction and related questions (2003); arXiv: math.PR/0307353.Google Scholar
  60. 60.
    D. B. Wilson, Generating random spanning trees more quickly than the cover time, in Proceedings of the Twenty-eighth Annual ACM Symposium on the Theory of Computing (Philadelphia, PA, 1996) (ACM, New York, 1996), pp. 296–303.Google Scholar
  61. 61.
    K. G. Wilson and J. Kogut, The renormalization group and the ε expansion, Phys. Rep. 12:75–199 (1974).Google Scholar

Copyright information

© Plenum Publishing Corporation 2004

Authors and Affiliations

  • Wouter Kager
    • 1
  • Bernard Nienhuis
    • 1
  1. 1.Institute for Theoretical PhysicsUniversity of AmsterdamAmsterdamThe Netherlands

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