Scaling limits of random curves at criticality

统计物理学中的许多数学模型的特点是，当参数达到一定值时，会发生急剧的质变。 这是相变的理想化，例如随着温度的变化，水从固体变成液体再变成气体。这些数学模型在参数的临界值处经常产生随机分形。 提议者将继续调查这些分形。 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，并且将对精细性质进行研究。三维空间的理论发展较少，主要目标是构造非平凡模型，二维空间的目标是研究自然分形参数化曲线上的Schramm-Loewner演化（SLE）。研究的技术问题包括曲线的保持器连续性（在自然参数化下）和SLE环测度的平稳性。后一个属性是需要构建一个共形协变措施SLE循环类似的布朗循环措施。 本文在三维空间中研究了几个问题：试图证明布朗运动存在唯一的时间环擦除，并将其极限描述为Laplacian随机游动，证明布朗运动相交指数的解析性，建立调和测度的多重分形谱;并找到一般的方法来给非平凡措施的连续，非自相交曲线的分形维数大于1。 无穷小（非标准）技术的使用将被认为是连续模型的替代品，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.