Geometry of random Loewner chains

本项目侧重于将概率和复杂分析技术应用于源于物理学的关于随机平面生长和晶格模型的缩放极限的数学问题。主要关注与Loewner微分方程相关的模型，特别是Schramm-Loewner演化（SLE）。提出了三个主要的研究方向。首先，PI打算主要从多重分形分析的角度研究SLE曲线的几乎确定的精细性质。其中一个主要问题涉及SLE船体边界上谐波测量的几乎确定的多重分形谱的性质，但PI也将研究，例如，曲线在尖端的缠绕以及SLE曲线与定义它的域边界碰撞的几何形状。所有这些问题都可以用产生SLE曲线的随机共形图的边界行为来表述。其次，PI打算研究SLE过程与相关离散模型和共形场理论（CFT）之间的严格联系。一个目标是从概率和解析的角度发展对CFT对象和代数结构的严格理解。最后，PI将研究Hastings-Levitov模型族，该模型族使用迭代随机保角映射来模拟平面聚集过程。一个特定的正则化将研究与尺度限制的理解和预测相变作为最终目标。离散概率格模型在物理学中被用来对一系列现象建模，它们提供了许多有趣的数学问题的来源。近年来，随机分形Schramm-Loewner演化曲线（SLE）的相关技术在这类模型的数学理解方面取得了很大进展，该曲线近似于模型中各阶段之间的界面。在该项目中，PI将发展对SLE曲线丰富的随机几何和多重分形结构的理解。除了这些结构的内在和基本的兴趣，以及它们与物理模型的紧密联系，SLE过程的普遍性表明，研究其几何特性的方法和见解将有助于研究其他随机分形。晶格模型的一种重要的物理方法是使用共形场论（CFTs），它（大致）是离散模型的“涂抹”连续版本。与非严格CFT预测的范围相比，严格的SLE机制仍然是有限的，从数学的角度来看，CFT和离散模型之间的许多联系仍然是神秘的。PI打算从概率和分析的角度对CFT的对象和结构进行严格的理解，特别是与SLE和离散模型本身的直接联系。一个相关的问题圈涉及聚集的模型，其中平面团簇是用来模拟重要的自然过程的，如扩散、有限聚集和粘性流体的流动。最近在了解与共形映射相关的随机增长模型方面取得的进展为PI提供了新的工具，PI将使用这些工具来研究所谓的Hastings-Levitov聚集模型族的版本。

This project focuses on the application of techniques from probability and complex analysis to mathematical problems concerning scaling limits of random planar growth and lattice model that originate in physics. The main focus is on models related to Loewner's differential equation, and in particular on the Schramm-Loewner evolution (SLE). Three principal directions of investigation are proposed. First, the PI intends to study almost sure fine properties of the SLE curves, primarily from the point of view of multifractal analysis. One of the main questions concerns the properties of the almost sure multifractal spectrum of harmonic measure on the boundary of the SLE hulls, but the PI will also investigate, e.g., the winding of the curve at the tip and the geometry of the SLE curve's collisions with the boundary of the domain where it is defined. All of these questions can be formulated in terms of the boundary behavior of the random conformal maps that generate the SLE curves. Secondly, the PI intends to study rigorous connections between the SLE processes and related discrete models and Conformal Field Theory (CFT). One goal is to develop the rigorous understanding of the objects and algebraic structures of CFT from probabilistic and analytic points of view. Finally, the PI will study the Hastings-Levitov family of models which uses iterated random conformal maps to model planar aggregation processes. A specific regularization will be studied with an understanding of scaling limits and the predicted phase transition as ultimate goals.Discrete probabilistic lattice models are used in physics to model a range of phenomena, and they provide a source of many interesting mathematical problems. Techniques related to the random fractal Schramm-Loewner evolution (SLE) curves, which approximate interfaces between phases in the models, have led to much progress in the mathematical understanding of such models in recent years. Within the project the PI will develop the understanding of the SLE curves' rich random geometric and multifractal structures. Beside the intrinsic and fundamental interest of these structures, and their strong connections with physical models, the universal nature of the SLE process suggests that methods and insights developed in the study of its geometric properties will be useful in the study of other random fractals. An important physics approach to lattice models uses Conformal Field Theories, CFTs, which (roughly) are ``smeared out'' continuum versions of the discrete models. The rigorous SLE machinery is still limited compared to the scope of the non-rigorous CFT predictions, and many connections between CFT and discrete models remain mysterious from a mathematical perspective. The PI intends to develop the rigorous understanding of the objects and structures of CFT from probabilistic and analytic points of view, in particular direct connections with SLE and the discrete models themselves. A related circle of questions concerns models of aggregation where planar clusters are grown to model important natural processes such as diffusion limited aggregation and flow of viscous fluid. Recent advances in the understanding of random growth models related to conformal maps has provided new tools which the PI will use to study versions of the so-called Hastings-Levitov family of aggregation models.