SLE Properties

系统性红斑狼疮特性

• 批准号: 0505751
• 负责人: Daniel Stroock
• 金额: $ 3.66万
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-09-01 至 2007-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0505751&HistoricalAwards=false
• 关键词: SLE; Properties

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.

SLE边界的Hausdorff维数上界的估计已经由S。Rohde和O.施拉姆虽然关于SLE迹的Hausdorff维数的定理是由V. Beffara宣布的，但当kappa 4时船体的边界仍然是一个开放的猜想。 为了得到SLE边界的Hausdorff维数的下界估计，采用了SLE反向流的归一化（前）Schwarzian导数的渐近行为。具有高阶项的SLE映射的正规化（预）Schwarzian导数是二阶矩服从Duplantier对偶的连续平方可积鞅，并且它们具有在双曲距离内指数衰减的相关性.这种方法允许人们对下界进行正式论证。马卡罗夫重对数定律使得调和测度与同指数函数相关的豪斯多夫测度相比较成为可能。自Schramm提出随机Loewner演化（Stochastic Loewner Evolution，SLE）以来，数学物理、概率论、复分析等领域的数学家和物理学家在这方面做了大量的工作，并在此基础上，提出了一种新的基于调和测度和Hausdorff测度的SLE重对数律估计方法。例如，SLE（6）被用来证明Mandelbrot猜想，即平面布朗路径的外边界应该具有分形维数4/3。 另一方面，统计物理学中的几种格点模型已经被证明或证明与某些SLE相对应。该项目为物理学家的预测提供数学分析，如Duplantier的对偶猜想，这是由共形场论的论点推导出来的。也可以建立其数学证明的方法。