Integrable Systems, Operator Determinants, and Probabilistic Models

本项目的重点是分析某些随机增长模型的极限律。所考虑的主要生长模型是相互作用的粒子系统，称为非对称简单排斥过程。这个模型是概率论和理论物理中被广泛研究的模型，因为它是表现非经典涨落的非平衡行为的最简单模型之一。在本项目中，我们将研究各种不同初始条件下的非对称简单排斥过程。这些方法使用了Bethe Anatz、Yang-Baxter方程的思想以及算子论和组合学的技术。将该模型推广到不止一个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”行为；事实上，将描述更大类别的增长模型中的波动。本项目涉及对各种初始构型的非对称简单排除过程中的当前波动的研究。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，因为它是将远离均衡的行为与非经典涨落结合在一起的最简单的模型之一。预计这些波动将具有一种新的普遍行为，其适用性类似于经典概率中著名的钟形曲线(高斯分布)。这一领域研究的一个长期目标是建立性质类似于经典中心极限定理的新的极限定律。这些新的普遍分布已经被应用于增长过程、人口遗传学和金融方面的各种问题。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.