Integrable Systems, Operator Determinants, and Probabilistic Models

这个项目的重点是分析某些随机增长模型的极限定律。 考虑的主要增长模型是相互作用的粒子系统称为不对称简单排斥过程。 该模型是概率论和理论物理中广泛研究的模型，因为它是表现出非经典波动的非平衡行为的最简单模型之一。 在这个项目中，我们将研究在各种不同初始条件下的非对称简单排它过程。 该方法使用的想法从贝特Anglesia，杨巴克斯特方程以及技术从算子理论和组合。 将模型推广到一个以上的物种也将是该项目的一部分。 预计这些极限法律将有一个“普遍”的行为，并会，事实上，描述的波动在一个更大的类growth models.This项目涉及的非对称简单的排斥过程中的各种初始配置的电流波动的研究。 这是一个一维晶格上相互作用粒子的模型。 由于它是将远离平衡态的行为与非经典涨落结合起来的最简单的模型之一，因此引起了数学家和物理学家的广泛关注。 这些涨落被认为具有一种新的普适性，类似于经典概率的著名钟形曲线（高斯分布）。 在这一领域的研究的一个长期目标是建立新的极限法律性质类似的经典中心极限定理。 这些新的普适分布已经被应用于增长过程、人口遗传学和金融学中的各种问题。 这个项目将把我们对波动的认识扩展到更广泛的增长模型。

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.