The Annals of Probability

The dimension of the SLE curves

Vincent Beffara

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Let γ be the curve generating a Schramm–Loewner Evolution (SLE) process, with parameter κ≥0. We prove that, with probability one, the Hausdorff dimension of γ is equal to Min(2, 1+κ/8).

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Ann. Probab. Volume 36, Number 4 (2008), 1421-1452.

First available: 29 July 2008

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Zentralblatt MATH identifier

Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 60G17: Sample path properties 28A80: Fractals [See also 37Fxx]

SLE Hausdorff dimension


Beffara, Vincent. The dimension of the SLE curves. The Annals of Probability 36 (2008), no. 4, 1421--1452. doi:10.1214/07-AOP364.

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  • [1] Bass, R. F. (1998). Diffusions and Elliptic Operators. Springer, New York.
  • [2] Beffara, V. (2004). Hausdorff dimensions for SLE6. Ann. Probab. 32 2606–2629.
  • [3] Camia, F. and Newman, C. M. (2006). The full scaling limit of two-dimensional critical percolation. Comm. Math. Phys. 268 1–38.
  • [4] Duplantier, B. (2000). Conformally invariant fractals and potential theory. Phys. Rev. Lett. 84 1363–1367.
  • [5] Friedrich, R. and Werner, W. (2002). Conformal fields, restriction properties, degenerate representations and SLE. C. R. Math. Acad. Sci. Paris 335 947–952.
  • [6] Kenyon, R. (2000). The asymptotic determinant of the discrete Laplacian. Acta Math. 185 239–286.
  • [7] Lawler, G. F. (1999). Geometric and fractal properties of Brownian motion and random walk paths in two and three dimensions. In Random Walks (Budapest, 1998). Bolyai Soc. Math. Stud. 9 219–258. János Bolyai Math. Soc., Budapest.
  • [8] Lawler, G. F., Schramm, O. and Werner, W. (2001). The dimension of the Brownian frontier is 4/3. Math. Res. Lett. 8 410–411.
  • [9] Lawler, G. F., Schramm, O. and Werner, W. (2001). Values of Brownian intersection exponents. II. Plane exponents. Acta Math. 187 275–308.
  • [10] Lawler, G. F., Schramm, O. and Werner, W. (2002). Sharp estimates for Brownian non-intersection probabilities. In In and Out of Equilibrium (Mambucaba, 2000). Progr. Probab. 51 113–131. Proceedings of the 4th Brazilian School of Probability. Birkhäuser, Boston.
  • [11] Lawler, G. F., Schramm, O. and Werner, W. (2003). Conformal restriction: The chordal case. J. Amer. Math. Soc. 16 917–955.
  • [12] Lawler, G. F., Schramm, O. and Werner, W. (2004). Conformal invariance of planar loop-erased random walks and uniform spanning trees. Ann. Probab. 32 939–995.
  • [13] Lawler, G. F., Schramm, O. and Werner, W. (2004). On the scaling limit of planar self-avoiding walk. In Fractal Geometry and Applications: A Jubilee of Benoît Mandelbrot, Part 2. Proc. Sympos. Pure Math. 72 339–364. Amer. Math. Soc., Providence, RI.
  • [14] Rohde, S. and Schramm, O. (2005). Basic properties of SLE. Ann. of Math. (2) 161 883–924.
  • [15] Schramm, O. (2000). Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math. 118 221–288.
  • [16] Schramm, O. and Sheffield, S. (2005). Harmonic explorer and its convergence to SLE4. Ann. Probab. 33 2127–2148.
  • [17] Smirnov, S. (2001). Critical percolation in the plane: Conformal invariance, Cardy’s formula, scaling limits. C. R. Acad. Sci. Paris Sér. I Math. 333 239–244.
  • [18] Smirnov, S. and Werner, W. (2001). Critical exponents for two-dimensional percolation. Math. Res. Lett. 8 729–744.
  • [19] Wilson, D. B. (1996). 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) 296–303. ACM, New York.