Walks in Random Media, Stochastic Growth and Pinning Effects.

概率论的基石是大数定律和中心极限定理的建立，它们的影响都超出了数学科学的范畴。粗略地说，具有有限二阶矩的n个独立同分布变量之和宏观上是n阶且有n^{1/2}阶的波动，服从高斯分布。在高斯波动的背后是总和与自变量集合的线性相关性。然而，自然界中的大多数现象都表现出对内在随机性的非线性依赖，挑战在于(I)理解传播随机性的非线性结构，(Ii)揭示这种机制的普遍特征。随机介质中的随机游动在统计物理中被广泛用于模拟这种现象。受到越来越高关注的两个这样的例子是(A)随机增长模型和(B)缺陷线上的钉扎模型。在情况(A)中，一个处理随机增长的界面。Kardar-Parisi-Zhang(KPZ)在80年代中期的非严格工作S通过预测这类模型表现出t^{1/3}起伏，建立了目前所知的KPZ普适性类的框架。最近的数学著作在特殊情况下将这类系统的涨落与来自随机矩阵理论的涨落联系在一起。我们的目标是建立一个严格的数学理论，通过研究这些模型的精确可解性质来解释这些波动的性质，将其与其他数学领域联系起来，并最终对其进行扰动，以揭示普遍现象。在情况(B)中，我们处理缺陷线附近的随机游动。我们的目标是理解与局域和非局域现象相关的相变。最近，与大偏差和粗粒化相关的技术被用来研究这种现象的相图。虽然已经取得了进展，但一些重要的问题仍然没有得到解决。我们建议通过构建这类模型的连续统极限来为该领域提供一条新的道路。通过这种方式，我们的目标是解决悬而未决的问题，并与KPZ现象建立深刻而新颖的联系。

A cornerstone of probability theory has been the establishment of the Law of Large Numbers and the Central Limit Theorem, both of them having impact beyond mathematical sciences. Roughly speaking, the sum of n independent, identically distributed variables with finite second moment is macroscopically of order n and has fluctuations of order n^{1/2}, obeying the Gaussian distribution. Underlying the Gaussian fluctuations is the linear dependence of the sum on the collection of the independent variables. However, most phenomena in nature exhibit a nonlinear dependence on the inherent randomness and the challenge is to (i) understand the nonlinear structure that propagates the randomness and (ii) reveal the universal features of this mechanism.Random walks in random media are widely used to model such phenomena in statistical physics. Two such instances that are receiving increasingly high attention are (A) stochastic growth models and (B) pinning models on defect lines. In case (A) one deals with a randomly growing interface. The non rigorous work of Kardar-Parisi-Zhang (KPZ) in the mid 80's set the framework of what is currently known as the KPZ universality class, by predicting that this class of models exhibits t^{1/3} fluctuations. More recent mathematical works have related, in special cases, the fluctuations of such systems to those coming from the theory of random matrices. Our goal is to build a rigorous mathematical theory that will explain the nature of these fluctuations by looking into the exactly solvable nature of these models, connect it to other mathematical fields and eventually perturb it in order to reveal universal phenomena.In case (B) one deals with a random walk in the vicinity of a defect line. The goal is to understand phase transitions related to localization and delocalization phenomena. Techniques related to large deviations and coarse graining have been used recently to study the phase diagrams of such phenomena. While progress has been made a number of important questions remain unresolved.We propose to provide a new path in the field through the construction of continuum limits of such models. In this way we aim to resolve the open questions and also make deep and novel connections to KPZ phenomena.