Integrable Systems, Operator Determinants, and Probabilistic Models

可积系统、算子决定因素和概率模型

• 批准号: 0906387
• 负责人: Craig Tracy
• 金额: $ 44.5万
• 依托单位: University of California-Davis
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-09-15 至 2013-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0906387&HistoricalAwards=false
• 关键词: Integrable; Systems; Operator; Determinants; Probabilistic

The focus of this project is the analysis of limit laws for certain stochastic growth models. The main growth model considered is the interacting particle system called the asymmetric simple exclusion process. This model is a widely studied model in probability theory and theoretical physics since it is one of the simplest models of nonequilibrium behavior that exhibits nonclassical fluctuations. In this project the asymmetric simple exclusion process will be studied for a variety of different initial conditions. The methods use ideas from Bethe Ansatz, Yang-Baxter equations as well as techniques from operator theory and combinatorics. Generalizations of the model to more than one species will also be part of the project. It is expected that these limit laws will have a "universal" behavior; and will, in fact, describe the fluctuations in a much larger class of growth models.This project involves the study of current fluctuations in the asymmetric simple exclusion process for a variety of initial configurations. This is a model of interacting particles on a one-dimensional lattice. The model has attracted wide attention from both mathematicians and physicists since it is one of the simplest models to incorporate far from equilibrium behavior with nonclassical fluctuations. These fluctuations are expected to have a new universal behavior similar in their applicability to the famous bell-shaped curve (the Gaussian distribution) of classical probability. A long-term goal of research in this area is the establishment of new limit laws similar in nature to the classical central limit theorem. Already these new universal distributions are being applied to various problems in growth processes, population genetics, and finance. This project will extend our knowledge of fluctuations to a much wider class of growth models.

该项目的重点是分析某些随机增长模型的极限定律。 考虑的主要增长模型是称为不对称简单排除过程的相互作用粒子系统。 该模型是概率论和理论物理学中广泛研究的模型，因为它是表现出非经典波动的非平衡行为的最简单模型之一。 在该项目中，将研究各种不同初始条件的不对称简单排除过程。 这些方法使用 Bethe Ansatz、Yang-Baxter 方程的思想以及算子理论和组合学的技术。 该模型对多个物种的推广也将是该项目的一部分。 预计这些限制法则将具有“普遍”行为；事实上，它将描述更大类别的增长模型中的波动。该项目涉及研究各种初始配置的不对称简单排除过程中的当前波动。 这是一维晶格上相互作用的粒子的模型。 该模型引起了数学家和物理学家的广泛关注，因为它是将远离平衡行为与非经典波动相结合的最简单模型之一。 这些波动预计将具有新的普遍行为，其适用性类似于经典概率的著名钟形曲线（高斯分布）。 该领域研究的长期目标是建立性质类似于经典中​​心极限定理的新极限定律。 这些新的普遍分布已经被应用于解决增长过程、人口遗传学和金融中的各种问题。 该项目将把我们对波动的知识扩展到更广泛的增长模型类别。