Geometry of random Loewner chains

随机 Loewner 链的几何结构

• 批准号: 1308476
• 负责人: Julien Dubedat
• 金额: $ 13.48万
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-09-15 至 2018-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1308476&HistoricalAwards=false
• 关键词: Geometry; random; Loewner; chains

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.

本项目主要研究将概率论和复分析的技术应用于与随机平面生长和晶格模型的标度极限有关的数学问题，这些问题源于物理学。主要重点是与Loewner微分方程相关的模型，特别是Schramm-Loewner演化（SLE）。提出了三个主要的调查方向。首先，PI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外壳边界上调和测度的几乎必然多重分形谱的性质，但PI也将研究，例如，曲线在尖端处的缠绕以及SLE曲线与其被定义的域的边界的碰撞的几何形状。所有这些问题都可以用生成SLE曲线的随机共形映射的边界行为来表述。其次，PI打算研究SLE过程与相关离散模型和共形场论（CFT）之间的严格联系。一个目标是从概率和分析的角度发展CFT的对象和代数结构的严格理解。最后，PI将研究Hastings-Levitov族模型，该模型使用迭代随机共形映射来模拟平面聚集过程。一个特定的正则化将与尺度限制的理解和预测的相变作为最终目标进行研究。离散概率晶格模型在物理学中用于模拟一系列现象，它们提供了许多有趣的数学问题的来源。与随机分形Schramm-Loewner演化（SLE）曲线，其中近似模型中的相位之间的接口，导致近年来在数学上的理解这样的模型取得了很大的进展。在项目中，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根本利益，以及它们与物理模型的密切联系之外，SLE过程的普遍性质表明，在研究其几何性质中开发的方法和见解将在其他随机分形的研究中有用。晶格模型的一个重要的物理方法是使用共形场论（Conformal Field Theories，CFTs），它（大致）是离散模型的连续版本。与非严格的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旨在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模型的聚集，平面集群的增长，以模拟重要的自然过程，如扩散限制聚集和流动的粘性流体。最近的进展，在了解随机增长模型有关的共形映射提供了新的工具，PI将用于研究版本的所谓的黑斯廷斯-Levitov家庭的聚集模型。