Boundary behavior of SLE. (English) Zbl 1198.60043
Summary: 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 [Ann. Math. (2) 161, No. 2, 883–924 (2005; Zbl 1081.60069)] to derive the conjectured sharp estimate for the Hölder exponent unless the parameter of SLE is 4.
We reexamine S. Rohde and O. Schramm’s derivative expectation [Ann. Math. (2) 161, No. 2, 883–924 (2005; Zbl 1081.60069)] to derive the conjectured sharp estimate for the Hölder exponent unless the parameter of SLE is 4.
MSC:
| 60K35 | Interacting random processes; statistical mechanics type models; percolation theory |
| 28A80 | Fractals |
| 30C55 | General theory of univalent and multivalent functions of one complex variable |
| 60J65 | Brownian motion |
Citations:
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