SLE Properties

SLE边界的Hausdorff维数上界的估计已经由S. Rohde和O. Schramm建立。虽然V. Beffara宣布了SLE迹线的Hausdorff维数定理，但kappa 4时船体的边界仍然是一个开放的猜想。为了估计SLE边界的Hausdorff维数的下界，利用了SLE逆向流的归一化（预）Schwarzian导数的渐近性质。具有高阶项的SLE映射的归一化（预）Schwarzian导数是二阶矩服从Duplantier对偶的连续平方可积鞅，它们具有在双曲距离中呈指数衰减的相关性。这种方法允许我们对下界作出正式的论证。马卡罗夫迭代对数定律使调和测度与对数微指数函数相关的豪斯多夫测度相比较成为可能。本项目的目标是得到SLE的Makarov迭代对数定律的精确估计，并将谐波测度与具有最佳可能测度函数的Hausdorff测度进行比较。自Schramm引入随机洛厄纳进化（Stochastic Loewner Evolution， SLE）以来，数学物理、概率论、复分析等各个领域的数学家和物理学家在这一领域做了大量的工作。例如，利用SLE(6)证明了Mandelbrot关于平面布朗路径外边界应具有4/3分形维数的猜想。另一方面，统计物理学中的几个晶格模型已经被证明或推测与一些SLE相对应。这个项目为物理学家的预测提供了数学分析，比如Duplantier的对偶猜想，这是由共形场论的论点推导出来的。它也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.