Integrable Systems, Operator Determinants, and Probabilistic Models

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性自旋链的极限定律：不对称简单排斥过程和量子力学海森堡-伊辛自旋链。我们的目标是使用可积系统，组合学，算子理论和渐近分析的技术来理解，并得出新的结果，这些probabilkistic模型。ASEP的基本问题之一是高度（或电流）波动的研究，特别是对出现的各种极限分布进行分类。证明了（KPZ普适性）波动对一大类增长模型是普适的。PI和克雷格A. Tracy在阶跃初始条件下建立了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，这将需要更好地理解组合恒等式。对于海森堡-伊辛模型的重点将是畴壁生长的动力学。该方法将需要一个新的使用贝特Anonymous，避免了尴尬的谱理论的海森堡-伊辛哈密尔顿。该项目将产生广泛的影响，在其他领域的数学和科学。来自随机矩阵理论和可积系统的思想和技术已经产生了影响，这样的不同学科，如概率，统计，生物统计，数论，凝聚态物理和工程。 特别是，ASEP模型在非平衡统计物理和生物系统中有应用。事实上，T（耳）ASEP是在20世纪60年代末作为核糖体沿着mRNA运动的模型引入的。对出现的极限定律的理解，特别是它们普遍存在的根本原因，将在这些领域具有很大的价值。这些普适分布最早是在数学文献中发现和计算的，后来发现了许多应用。人们完全可以预期，这些法律在ASEP的阐明将进一步影响应用领域-不仅创造新的技术来解决长期存在的问题，在上述领域，但也提供了新的和意想不到的应用。

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.