×

zbMATH — the first resource for mathematics

Hausdorff dimensions for SLE\(_6\). (English) Zbl 1055.60036
The author proves that the Hausdorff dimension of the trace of SLE\(_6\) is almost surely \(7/4\) and the dimension of its trace is \(4/3\). Using this, he gives a more direct deviation of the result of G. F. Lawler, O. Schramm and W. Werner [Math. Res. Lett. 8, No. 1–2, 13–23 (2001; Zbl 0993.60085)] that the dimension of the planar Brownian frontier is \(4/3\). The author also proves that, for all \(\kappa<8\), the SLE\(_\kappa\) trace has cut-points.

MSC:
60G17 Sample path properties
28A80 Fractals
60J55 Local time and additive functionals
60K35 Interacting random processes; statistical mechanics type models; percolation theory
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Bass, R. F. (1998). Diffusions and Elliptic Operators . Springer, New York. · Zbl 0914.60009
[2] Beffara, V. (2002). \(\SLE_\kappa\) when \(\kappa\to0\) or \(\kappa\to\infty\). Unpublished manuscript.
[3] Beffara, V. (2003). On conformally invariant subsets of the planar Brownian curve. Ann. Inst. H. Poincaré Probab. Statist. 39 793–821. · Zbl 1021.60064
[4] Kaufman, R. (1969). Une propriété métrique du mouvement Brownien. C. R. Acad. Sci. Paris Sér. A 268 727–728. · Zbl 0174.21401
[5] Lawler, G. F. (1999). Geometric and fractal properties of Brownian motion and random walk paths in two and three dimensions. Bolyai Soc. Math. Stud. 9 219–258. · Zbl 0955.60076
[6] Lawler, G. F., Schramm, O. and Werner, W. (2001). The dimension of the Brownian frontier is \(4/3\). Math. Rev. Lett. 8 13–24. · Zbl 0993.60085
[7] Lawler, G. F., Schramm, O. and Werner, W. (2001). Values of Brownian intersection exponents I: Half-plane exponents. Acta Math. 187 237–273. · Zbl 1005.60097
[8] Lawler, G. F., Schramm, O. and Werner, W. (2001). Values of Brownian intersection exponents II: Plane exponents. Acta Math. 187 275–308. · Zbl 0993.60083
[9] Lawler, G. F., Schramm, O. and Werner, W. (2002). Values of Brownian intersection exponents III: Two-sided exponents. Ann. Inst. H. Poincaré Probab. Statist. 38 109–123. · Zbl 1006.60075
[10] Lawler, G. F., Schramm, O. and Werner, W. (2002). Analyticity of intersection exponents for planar Brownian motion. Acta Math. 188 179–201. · Zbl 1024.60033
[11] Lawler, G. F., Schramm, O. and Werner, W. (2003). Conformal restriction: The chordal case. J. Amer. Math. Soc. 16 917–955. · Zbl 1030.60096
[12] Lawler, G. F., Schramm, O. and Werner, W. (2004). On the scaling limit of planar self-avoiding walks. In Fractal Geometry and Application . · Zbl 1069.60089
[13] Lawler, G. F., Schramm, O. and Werner, W. (2004). Conformal invariance of planar loop-erased random walks and uniform spanning trees. Ann. Probab. · Zbl 1126.82011
[14] Pommerenke, C. (1992). Boundary Behaviour of Conformal Maps . Springer, New York. · Zbl 0762.30001
[15] Rohde, S. and Schramm, O. (2001). Basic properties of \(\SLE\). Ann. of Math. · Zbl 1081.60069
[16] Schramm, O. (2000). Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math. 118 221–288. · Zbl 0968.60093
[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. · Zbl 0985.60090
[18] Smirnov, S. and Werner, W. (2001). Critical exponents for two-dimensional percolation. Math. Res. Lett. 8 729–744. · Zbl 1009.60087
[19] Werner, W. (2002). Random planar curves and Schramm–Loewner evolutions. École d ’ Éte de Probabilités de Saint-Flour XXXII . Lecture Notes in Math. 1840 107–195. Springer, New York. · Zbl 1057.60078
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.