Integrable Systems, Operator Determinants, and Probabilistic Models

该项目的主要活动是分析某些随机生长模型/相互作用粒子系统和磁自旋链的极限定律：不对称简单不相容过程（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的技术来理解这些概率模型，并推导出新的结果。ASEP的一个基本问题是对高度（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。我们推测（KPZ普适性）波动对于一大类增长模型是普适性的。PI和Craig a . Tracy建立了阶跃初始条件ASEP的KPZ普适性，这一结果需要一些新的组合恒等式。该项目提出了其他初始条件的研究，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，重点将放在畴壁生长的动力学上。该方法将需要贝特·安萨茨的一种新用法，以避免海森堡-伊辛哈密顿量的尴尬谱理论。这个项目将对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已经对诸如概率论、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(total)ASEP是在20世纪60年代末作为核糖体沿mRNA运动的模型引入的。对出现的极限定律的理解，特别是对它们普遍存在的潜在原因的理解，在这些领域将具有很大的价值。这些普适分布最初是在数学文献中被发现和计算出来的，并且已经有了许多应用。完全可以预期，ASEP中这些法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中长期存在的问题，而且还提供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.