SLE Properties

S.Rohde和O.Schramm已经建立了SLE边界的Hausdorff维数的上界估计。而V.Beffara提出了关于SLE迹的Hausdorff维数的一个定理，而kappa 4时船体的边界仍然是一个公开的猜想。利用归一化(前)Schwarzian导数的渐近行为，得到了SLE边界的Hausdorff维数的下界估计。具有高阶项的SLE映射的归一化(准)Schwarzian导数是具有二阶矩的连续平方可积鞅，它们具有在双曲距离上指数衰减的相关性。这种方法允许人们对下限进行正式的论证。马卡洛夫重对数律使调和度量与与对数指数函数相关的Hausdorff度量的比较成为可能。该项目的目的是得到马卡洛夫重对数律的精确估计，并将调和测度与具有最佳可能测度函数的Hausdorff测度进行比较。自从Schramm提出随机Loewner进化(SLE)以来，数学家和物理学家在数学物理、概率论和复分析等领域做了大量的工作。例如，SLE(6)被用来证明Mandelbrot的猜想，即平面布朗路径的外边界应该具有4/3的分维。另一方面，统计物理学中的几个晶格模型已经被证明或猜测与一些SLE相对应。这个项目为物理学家的预测提供了数学分析，如Duplan tier的对偶猜想，它是由共形场论的论点推导出来的。它还可以建立方法来接近他们的数学证明。

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.