Journal of the American Mathematical Society

Author(s): Nam-Gyu Kang
Journal: J. Amer. Math. Soc. 20 (2007), 185-210.
MSC (2000): Primary 30C45, 60K35; Secondary 28A80, 60J65
Posted: August 28, 2006
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Abstract: We show that the normalized (pre-)Schwarzian derivative of SLE, after we subtract a negligible term, is a complex BMO martingale. Its BMO norm gives strong evidence for Duplantier's duality conjecture. We also show that it has correlations that decay exponentially in the hyperbolic distance.

We reexamine S. Rohde and O. Schramm's derivative expectation to derive the conjectured sharp estimate for the Hölder exponent unless the parameter of SLE is 4.

1.: V. Beffara, The dimension of the SLE curves, preprint, math.PR/0211322 (2002).
2.: I. Binder, Rotational spectrum of planar domains, Ph.D. thesis, Caltech, 1997.
3.: C. J. Bishop and P. W. Jones, Harmonic measure, $L\sp 2$ estimates and the Schwarzian derivative, J. Anal. Math. 62 (1994), 77-113. MR 1269200 (95f:30034)
4.: -, Wiggly sets and limit sets, Ark. Mat. 35 (1997), no. 2, 201-224. MR 1478778 (99f:30066)
5.: C. J. Bishop, P. W. Jones, R. Pemantle, and Y. Peres, The dimension of the Brownian frontier is greater than , J. Funct. Anal. 143 (1997), no. 2, 309-336. MR 1428819 (97k:60105)
6.: L. de Branges, A proof of the Bieberbach conjecture, Acta Math. 154 (1985), no. 1, 137-152. MR 772434 (86h:30026)
7.: B. Duplantier, Conformally invariant fractals and potential theory, Phys. Rev. Lett. 84 (2000), no. 7, 1363-1367. MR 1740371 (2001c:82040)
8.: -, Conformal fractal geometry and boundary quantum gravity, Fractal geometry and applications: a jubilee of Benoît Mandelbrot, Part 2: Multifractals, probability and statistical mechanics, applications (Providence), Proc. Sympos. Pure Math., vol. 72, Amer. Math. Soc., 2004, pp. 365-482. MR 2112128 (2005m:82057)
9.: B. Duplantier and I. Binder, Harmonic measure and winding of conformally invariant curves, Phys. Rev. Lett. 89 (2002), 264101.
10.: R. Durrett, Brownian motion and martingales in analysis, Wadsworth, Belmont, 1984. MR 750829 (87a:60054)
11.: A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher transcendental functions. Vol. I., McGraw-Hill Book Company, Inc., New York-Toronto-London, 1953, based in part on notes left by Harry Bateman. MR 698779 (84h:33001a)
12.: G. B. Folland, Fourier analysis and its applications, Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, 1992. MR 1145236 (93f:42001)
13.: J. Garnett and D. E. Marshall, Harmonic measure, Cambridge University Press, Cambridge, 2005. MR 2150803 (2006g:31002)
14.: G. M. Goluzin, Geometric theory of functions of a complex variable, Translations of Mathematical Monographs, vol. 26, Amer. Math. Soc., Providence, 1969. MR 0247039 (40:308)
15.: J. Graczyk and P. W. Jones, Dimension of the boundary of quasiconformal Siegel disks, Invent. Math. 148 (2002), no. 3, 465-493. MR 1908057 (2003c:37063)
16.: F. John and L. Nirenberg, On functions of bounded mean oscillation, Comm. Pure Appl. Math. 14 (1961), 415-426. MR 0131498 (24:A1348)
17.: P. W. Jones, Rectifiable sets and the traveling salesman problem, Invent. Math. 102 (1990), no. 1, 1-15. MR 1069238 (91i:26016)
18.: I. Karatzas and S. Shreve, Brownian motion and stochastic calculus, Springer-Verlag, Berlin, 1991. MR 1121940 (92h:60127)
19.: G. F. Lawler, The dimension of the frontier of planar Brownian motion, Electron. Comm. Probab. 1 (1996), 29-47. MR 1386292 (97g:60110)
20.: -, Conformally invariant processes in the plane, Mathematical Surveys and Monographs, vol. 114, Amer. Math. Soc., Providence, 2005. MR 2129588 (2006i:60003)
21.: G. F. Lawler, O. Schramm, and W. Werner, The dimension of the Brownian frontier is 4/3, Math. Res. Lett. 8 (2001), no. 4, 401-411. MR 1849257 (2003a:60127b)
22.: -, Values of Brownian intersection exponents I: Half plane exponents, Acta Math. 187 (2001), no. 2, 237-273. MR 1879850 (2002m:60159a)
23.: -, Values of Brownian intersection exponents II: Plane exponents, Acta Math. 187 (2001), no. 2, 275-308. MR 1879851 (2002m:60159b)
24.: -, Analyticity of intersection exponents for planar Brownian motion, Acta Math. 189 (2002), no. 2, 179-201. MR 1961197 (2003m:60231)
25.: -, Values of Brownian intersection exponents III: Two sided exponents, Ann. Inst. H. Poincaré Probab. Statist. 38 (2002), no. 1, 109-123. MR 1899232 (2003d:60163)
26.: -, Conformal invariance of planar loop-erased random walks and uniform spanning trees, Ann. Probab. 32 (2004), no. 1B, 939-995. MR 2044671 (2005f:82043)
27.: -, On the scaling limit of planar self-avoiding walk, Fractal geometry and applications: a jubilee of Benoît Mandelbrot, Part 2: Multifractals, probability and statistical mechanics, applications (Providence), Proc. Sympos. Pure Math., vol. 72, Amer. Math. Soc., 2004, pp. 339-364. MR 2112127 (2006d:82033)
28.: J. Lind, Hölder regularity for the SLE trace, preprint (2005).
29.: Ch. Pommerenke, Boundary behaviour of conformal maps, Springer-Verlag, Berlin, 1992. MR 1217706 (95b:30008)
30.: J. Ratcliffe, Foundations of hyperbolic manifolds, Graduate Texts in Mathematics, vol. 149, Springer-Verlag, New York, 1994. MR 1299730 (95j:57011)
31.: S. Rohde and O. Schramm, Basic properties of SLE, Ann. of Math. 161 (2005), no. 2, 883-924. MR 2153402 (2006f:60093)
32.: B. Schmuland and W. Sun, A central limit theorem and law of the iterated logarithm for a random field with exponential decay of correlations, Canad. J. Math. 56 (2004), no. 1, 209-224. MR 2031129 (2005f:60122)
33.: O. Schramm, Scaling limits of loop-erased random walks and uniform spanning trees, Israel J. Math. 118 (2000), 221-288. MR 1776084 (2001m:60227)
34.: O. Schramm and S. Sheffield, Harmonic explorer and its convergence to ${\rm SLE}\sb 4$ , Ann. Probab. 33 (2005), no. 6, 2127-2148. MR 2184093 (2006i:60013)
35.: S. Smirnov, Critical percolation in the plane: conformal invariance, Cardy's formula, scaling limits, C. R. Acad. Sci. Paris Sér. I. Math. 333 (2001), no. 3, 239-244. MR 1851632 (2002f:60193)
36.: E. M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton University Press, Princeton, 1993. MR 1232192 (95c:42002)
37.: W. Werner, Random planar curves and Schramm-Loewner evolutions, Lectures on probability theory and statistics: Ecole d'Eté de Probabilités de Saint-Flour XXXII--2002, Lecture Notes in Mathematics, vol. 1840, Springer-Verlag, Berlin, 2004, pp. 107-195. MR 2079672 (2005m:60020)

Retrieve articles in Journal of the American Mathematical Society with MSC (2000): 30C45, 60K35, 28A80, 60J65

Nam-Gyu Kang
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
Email: kang@math.mit.edu

DOI: 10.1090/S0894-0347-06-00547-9
PII: S 0894-0347(06)00547-9
Keywords: SLE, Schwarzian derivatives, H\"{o}lder continuity, Duplantier's duality conjecture
Received by editor(s): January 31, 2005
Posted: August 28, 2006
Additional Notes: This research was partially conducted during the period when the author was employed by the Clay Mathematical Institute as a Liftoff Fellow. The author is partially supported by NSF grant DMS 05-05751.
Copyright of article: Copyright 2006, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

Kang, Nam-Gyu, On the quantitative boundary behavior of SLE, Universality and renormalization, Fields Inst. Commun., vol. 50, Amer. Math. Soc., Providence, RI, 2007, pp. 185--198. MR 2310305

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