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: 0854934
• 负责人: Harold Widom
• 金额: $ 30.03万
• 依托单位: University of California-Santa Cruz
• 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: 2009-06-01 至 2015-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0854934&HistoricalAwards=false
• 关键词: Integrable; Systems; Operator; Determinants; Probabilistic

The main activity of the project is the analysis of limit laws for certain stochastic growth models/interacting particle systems and magnetic spin chains: the asymmetric simple exclusion process (ASEP) and the quantum-mechanical Heisenberg-Ising spin chain. The objective is to use the techniques of integrable systems, combinatorics, operator theory, and asymptotic analysis to understand, and to derive new results for, these probabilkistic models. One of the basic problems of ASEP is the study of height (or current) fluctuations, in particular to classify various limiting distributions that arise. It is conjectured (KPZ universality) that the fluctuations are universal for a large class of growth models. The PI and Craig A. Tracy established KPZ universality for ASEP with step initial condition, a result that required some new combinatorial identities. The project proposes the study of other initial conditions, in particular Bernoulli initial condition, which would require a better understanding of the combinatorial identities. For the Heisenberg-Ising model the focus will be on the dynamics of domain wall growth. The method will entail a novel use of the Bethe Ansatz that avoids the awkward spectral theory of the Heisenberg-Ising Hamiltonian.The project would have broad impact in other areas of mathematics and science.The ideas and techniques coming from random matrix theory and integrable systems have had an impact on such diverse subjects as probability, statistics, biostatistics, number theory , condensed matter physics, and engineering. In particular, the ASEP model has applications in nonequilibrium statistical physics and biological systems. In fact, the T(otally)ASEP was introduced in the late 1960s as a model of ribosome motion along mRNA. The understanding of the limit laws that arise, in particular the underlying reason for their ubiquitous occurrence, will have great value in these areas. These universal distributions were first discovered and computed in the mathematical literature and have since found many applications. It is fully expected that elucidation of these laws in ASEP will further impact the applied areas -- not only to create new techniques to solve long-standing problems in the above-mentioned fields but also to proviude new and unexpected applications.

该项目的主要活动是分析某些随机增长模型/相互作用粒子系统和磁自旋链的极限定律：不对称简单排除过程（ASEP）和量子力学海森堡-伊辛自旋链。目标是使用可积系统、组合学、算子理论和渐近分析技术来理解这些概率模型并得出新结果。 ASEP 的基本问题之一是研究高度（或电流）波动，特别是对出现的各种极限分布进行分类。据推测（KPZ 普遍性），波动对于一大类增长模型来说是普遍的。 PI 和 Craig A. Tracy 在步骤初始条件下建立了 ASEP 的 KPZ 普适性，这一结果需要一些新的组合恒等式。该项目建议研究其他初始条件，特别是伯努利初始条件，这需要更好地理解组合恒等式。对于海森堡-伊辛模型，重点将放在磁畴壁生长的动态上。该方法将需要对 Bethe Ansatz 进行新颖的使用，从而避免海森堡-伊辛哈密顿量的尴尬谱理论。该项目将在数学和科学的其他领域产生广泛的影响。来自随机矩阵理论和可积系统的思想和技术对概率、统计学、生物统计学、数论、凝聚态物理和工程学等不同学科产生了影响。 特别是，ASEP 模型在非平衡统计物理和生物系统中具有应用。事实上，T(otally)ASEP 是在 20 世纪 60 年代末引入的，作为核糖体沿 mRNA 运动的模型。了解所出现的极限定律，特别是其普遍存在的根本原因，将在这些领域具有巨大的价值。这些通用分布首先在数学文献中发现和计算，此后发现了许多应用。完全可以预期，ASEP 对这些定律的阐明将进一步影响应用领域——不仅创造新技术来解决上述领域长期存在的问题，而且还提供新的和意想不到的应用。