Scaling limits of random curves at criticality

统计物理学中的许多数学模型的特点是，当一个参数达到某一值时，会有一个急剧的质变。这是相变的理想化，例如随着温度的变化，水从固体到液体再到气体的变化。这些数学模型在参数的临界值经常产生随机的分形图。提交人将继续调查这些分形图。这一理论在二维中得到了很好的发展，其中系统在保角变换下是不变的，研究将是关于精细性质的。三维空间的理论还不是很成熟，主要目的是构造非平凡模型，而二维空间的目标则是研究Schramm-Loewner演化(SLE)作为曲线的一种度量，而曲线的参数化用自然的分形参数表示。要研究的技术问题包括曲线的Holder连续性(在自然参数化下)和SLE环测的平稳性。后一性质是构造SLE环上类似于Brown环的共形协变测度所必需的。在三维空间中有几个问题将被研究：试图证明布朗运动存在唯一的按时间顺序的环-擦除，并将极限描述为拉普拉斯随机游动；证明布朗运动的相交指数的解析性并建立调和测度的多重分形谱；以及寻找在分维大于1的连续的、非自相交的曲线上给出非平凡度量的一般方法。无限小(非标准)技术的使用将被认为是对象的连续介质模型的替代，例如高斯自由场的指数、均匀生成树和渗流。

Many mathematical models from statistical physics are characterized by the fact that there is a sharp qualitative change when a parameter reaches a certain value. This is an idealization of phase transitions such as the change of water from solid to liquid to gas as temperature varies. These mathematical models at critical values of the parameter often produce random fractals. The proposer will continue investigation of these fractals. The theory is well developed in two dimensions where the systems are invariant under conformal transformation and the investigation will be on fine properties. The theory in three dimensions is less developed and the main goal is to construct nontrivial models.More specifically, the goal in two dimensions is to study the Schramm-Loewner evolution (SLE) as a measure on curves parametrized with the natural fractal parametrization. Technical questions to study include the Holder continuity of the curves (under natural parametrization) and the stationary of the SLE loop measure. The latter property is needed to construct a conformally covariant measure on SLE loops analogous to the Brownian loop measure. There are several questions in three dimensions that will be investigated: trying to show that there is a unique chronological loop-erasure of Brownian motion and to describe the limit as a Laplacian random walk; proving analyticity of the intersection exponent for Brownian motion and establishing the multifractal spectrum for harmonic measure; and finding general methods to give nontrivial measures on continuous, non-self-intersecting curves of fractal dimension greater than one. The use of infinitesimal (non-standard) techniques will be considered as an alternative to continuum models for objects such as exponentials of Gaussian free fields, uniform spanning trees, and percolation.