Beffara, Vincent The dimension of the SLE curves. (English) Zbl 1165.60007 Ann. Probab. 36, No. 4, 1421-1452 (2008). Let the random curve \(\gamma:[0,\infty)\to \overline{\mathbb{H}}\) be a trace of the chordal \(SLE_{\kappa}\) where \(\overline{\mathbb{H}}\) is a closure of the upper-half plane \(\mathbb{H}\) (for details concerning the Schramm–Loewner Evolution, i.e. a family of maps \(\{g_t,t\geq 0\}\) called \(SLE_{\kappa}\) and introduced by the following PDE: \[ \partial_tg_t(z) = \frac{2}{g_t(z) - \sqrt{\kappa} B_t},\;\;g_0(z)=z, \] where \((B_t)\) is a standard Brownian motion and \(\kappa\) is a positive parameter, see, e.g., [S. Rohde, O. Schramm, Ann. Math, (2) 161, 883–924 (2005; Zbl 1081.60069)]. In the above mentioned paper the authors proved that the Hausdorff dimension of the \(SLE_{\kappa}\) trace is not larger than \(1+\frac{\kappa}{8}\) when \(\kappa \leq 8\), and they conjectured the sharpness of this bound. The main result of the present paper provides the proof of this conjecture. Namely, almost surely one has \(\dim_H(\mathcal{H}) = 2 \wedge (1+ \frac{\kappa}{8})\) where \(\mathcal{H}= \gamma([0,\infty))\). This result was known for \(\kappa \geq 8\) because the curve is then space-filling. Reviewer: Alexander V. Bulinski (Moskva) Cited in 1 ReviewCited in 108 Documents MSC: 60D05 Geometric probability and stochastic geometry 60G17 Sample path properties 28A80 Fractals Keywords:Schramm–Loewner Evolution; trace of the chordal \(SLE_{\kappa}\); Hausdorff dimension. Citations:Zbl 1081.60069 × Cite Format Result Cite Review PDF Full Text: DOI arXiv References: [1] Bass, R. F. (1998). Diffusions and Elliptic Operators . Springer, New York. · Zbl 0914.60009 · doi:10.1007/b97611 [2] Beffara, V. (2004). Hausdorff dimensions for SLE 6 . Ann. Probab. 32 2606-2629. · Zbl 1055.60036 · doi:10.1214/009117904000000072 [3] Camia, F. and Newman, C. M. (2006). The full scaling limit of two-dimensional critical percolation. Comm. Math. Phys. 268 1-38. · Zbl 1117.60086 · doi:10.1007/s00220-006-0086-1 [4] Duplantier, B. (2000). Conformally invariant fractals and potential theory. Phys. Rev. Lett. 84 1363-1367. · Zbl 1042.82577 · doi:10.1103/PhysRevLett.84.1363 [5] Friedrich, R. and Werner, W. (2002). Conformal fields, restriction properties, degenerate representations and SLE. C. R. Math. Acad. Sci. Paris 335 947-952. · Zbl 1101.81095 · doi:10.1016/S1631-073X(02)02581-5 [6] Kenyon, R. (2000). The asymptotic determinant of the discrete Laplacian. Acta Math. 185 239-286. · Zbl 0982.05013 · doi:10.1007/BF02392811 [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. · Zbl 0955.60076 [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. · Zbl 1114.60316 [9] 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 · doi:10.1007/BF02392619 [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. · Zbl 1011.60062 [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 · doi:10.1090/S0894-0347-03-00430-2 [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. · Zbl 1126.82011 · doi:10.1214/aop/1079021469 [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. · Zbl 1069.60089 [14] Rohde, S. and Schramm, O. (2005). Basic properties of SLE. Ann. of Math. ( 2 ) 161 883-924. · Zbl 1081.60069 · doi:10.4007/annals.2005.161.883 [15] Schramm, O. (2000). Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math. 118 221-288. · Zbl 0968.60093 · doi:10.1007/BF02803524 [16] Schramm, O. and Sheffield, S. (2005). Harmonic explorer and its convergence to SLE 4 . Ann. Probab. 33 2127-2148. · Zbl 1095.60007 · doi:10.1214/009117905000000477 [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 · doi:10.1016/S0764-4442(01)01991-7 [18] Smirnov, S. and Werner, W. (2001). Critical exponents for two-dimensional percolation. Math. Res. Lett. 8 729-744. · Zbl 1009.60087 · doi:10.4310/MRL.2001.v8.n6.a4 [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. · Zbl 0946.60070 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.