Integrable Systems, Operator Determinants, and Probabilistic Models

本课题的重点是对某些随机增长模型的极限规律进行分析。考虑的主要生长模型是称为不对称简单排斥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被广泛研究的模型，因为它是表现非经典波动的非平衡行为的最简单模型之一。本课题将研究不同初始条件下的非对称简单不相容过程。这些方法使用了Bethe Ansatz， Yang-Baxter方程的思想以及算子理论和组合学的技术。将该模型推广到一个以上的物种也将是该项目的一部分。预计这些极限定律将具有“普遍”的行为；事实上，我们将描述更大的一类增长模型中的波动。本项目涉及对各种初始构型的非对称简单排斥过程中的电流波动进行研究。这是一维晶格中粒子相互作用的模型。该模型是包含非经典波动和非平衡行为的最简单模型之一，因此引起了数学家和物理学家的广泛关注。预计这些波动将具有一种新的普遍行为，类似于经典概率中著名的钟形曲线（高斯分布）的适用性。这一领域的长期研究目标是建立与经典中心极限定理性质相似的新极限定律。这些新的普遍分布已经被应用于生长过程、人口遗传学和金融等各种问题。这个项目将把我们关于波动的知识扩展到更广泛的增长模型类别。

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.