SLE Properties

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The estimate for the upper bound of the Hausdorff dimension of the SLE boundary is already established by S. Rohde and O. Schramm. While a theorem on the Hausdorff dimension of the SLE trace was announced by V. Beffara, the boundary of the hull when kappa 4 remains an open conjecture. To get an estimate on the lower bound for the Hausdorff dimension of the SLE boundary, the asymptotic behavior of normalized (pre-)Schwarzian derivatives for the SLE backward flows is employed. The normalized (pre-)Schwarzian derivatives of SLE maps with higher order terms are continuous square integrable martingales with second moment obeying the Duplantier duality and they have correlations that decay exponentially in the hyperbolic distance. This method allows one to make a formal argument for a lower bound. Makarov's law of iterated logarithm makes it possible to compare harmonic measure to a Hausdorff measure associated with a logarithmico-exponential function. The goal of this project is to get the exact estimate for Makarov's law of iterated logarithm for SLE and compare harmonic measure to a Hausdorff measure with a best possible measure function.Since Stochastic Loewner Evolution (SLE) was introduced by Schramm, a lot of works have been done in this area by mathematicians and physicists in various areas like mathematical physics, probability theory and complex analysis. For example, SLE(6) was used to prove Mandelbrot's conjecture that the outer boundary of a planar Brownian path should have fractal dimension 4/3. On the other hand, several lattice models from statistical physics have been shown or conjectured to correspond to some SLE's. This project provides mathematical analysis for physicists' predictions like Duplantier's duality conjecture, which is derived by arguments from conformal field theory. It may also build up the methods to approach their mathematical proofs.