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In this paper we study the multifractal structure of Schramm’s SLE curves. We derive the values of the (average) spectrum of harmonic measure and prove Duplantier’s prediction for the multifractal spectrum of SLE curves. The spectrum can also be used to derive estimates of the dimension, Hölder exponent and other geometrical quantities. The SLE curves provide perhaps the only example of sets where the spectrum is non-trivial yet exactly computable. | Commun. Math. Phys. 290 577-595 2009 Digital Object Identifier DOI 10.1007 s00220-009-0864-7 Communications in Mathematical Physics Harmonic Measure and SLE D. Beliaev1 2 S. Smirnov3 1 Department of Mathematics Fine Hall Princeton University Princeton NJ 08544 USA. E-mail dbeliaev@math.princeton.edu 2 IAS Princeton NJ 08544 USA 3 Section de Mathématiques Université de Genève 2-4 rue du Lièvre CH-1211 Genève 4 Switzerland. E-mail Stanislav.Smirnov@math.unige.ch Received 27 March 2008 Accepted 10 April 2009 Published online 7 July 2009 - Springer-Verlag 2009 Abstract In this paper we study the multifractal structure of Schramm s SLE curves. We derive the values of the average spectrum of harmonic measure and prove Duplantier s prediction for the multifractal spectrum of SLE curves. The spectrum can also be used to derive estimates of the dimension Holder exponent and other geometrical quantities. The SLE curves provide perhaps the only example of sets where the spectrum is non-trivial yet exactly computable. 1. Introduction The motivation for this paper is twofold to study multifractal spectrum of the harmonic measure and to better describe the geometry of Schramm s SLE curves see Sects. 1.1 and 1.2 for brief introductions to the respective subjects . Our main result is the following theorem in which we rigorously compute the average spectrum of harmonic measure on domains bounded by SLE curves see below for precise definitions . Theorem 1. The average integral means spectrum j3 t ofSLEK is equal to 4 k y 4 k 2 8tk t k 4k 3k t - T 4 k 4 k 7 4 k 2 8tk t 4k 3k 1 8 - - 3 4 k 2 32k t 4 k 2 t . 16k 32k 578 D. Beliaev S. Smirnov The average integral means spectrum ft t of the bulk of SLE see definition below is equal to 5 t 4 k 4 K -J 4 k 2 8tK 4k 3 4 k 2 t - 32k 3 4 k 2 t 32k 4 k 2 16k Several results can be easily derived from this theorem dimension estimates of the boundary of SLEK hulls Holder continuity of SLEK Riemann maps Holder continuity of SLEk trace and more. We
