Scaling limits of random curves at criticality

统计物理中的许多数学模型的特点是，当参数达到某一值时，会发生急剧的质变。这是一种理想化的相变，例如水随着温度的变化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处经常产生随机分形。提案人将继续研究这些分形。该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，并且将研究精细性质。三维的理论发展较少，主要目标是构建非平凡模型。更具体地说，在二维的目标是研究Schramm-Loewner演化（SLE）作为曲线参数化与自然分形参数化的度量。需要研究的技术问题包括曲线的Holder连续性（在自然参数化条件下）和SLE环路测量的平稳性。后一种性质需要在SLE环上构造一个类似于布朗环测度的共形协变测度。在三维空间中有几个问题需要研究：试图证明布朗运动存在一个独特的时间顺序循环消除，并将其极限描述为拉普拉斯随机漫步；证明了布朗运动相交指数的可解析性，建立了谐波测度的多重分形谱；寻找分形维数大于1的连续非自交曲线的非平凡测度的一般方法。使用无穷小（非标准）技术将被认为是连续体模型的替代品，例如高斯自由场的指数，均匀生成树和渗透。

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.