Structures and universalities around the Kardar-Parisi-Zhang equation

Kardar，Parisi和Zhang在20世纪80年代提出了一大类随机生长的界面呈现普适涨落的非线性随机偏微分方程，现在称为Kardar-Parisi-Zhang或KPZ方程。这一类中的模型展示了三种基本机制：作为界面陡度函数的增长，由拉普拉斯模型建模的平滑效果，以及由白噪声建模的局部随机性。这类物理系统的例子有液体在多孔介质中的渗流、细菌菌落的生长、一维交通或液体系统中的电流、液晶等。值得注意的是，这种随机界面的涨落是由指数和分布决定的，而这些指数和分布与经典的中心极限定理所给出的预测不同。在第一个维度，令人惊讶的是，它们与随机矩阵理论中出现的定律有关，因为这首先是由Baik-Deift-Johansson的工作展示的，随后是一系列活动，这些活动建立了“行列式过程”的框架。最近几年发生了令人振奋的新发展，使人们迈出了超越行列式模型的普遍性的重要第一步。在第二维，由于控制指数和分布未知，甚至二维KPZ的含义也没有设定到位，情况要落后得多。该项目的目标有两个：a.通过建立一个稳健的框架来研究不确定系统的涨落，攻击关于多点关联的未决猜想，以及探索普适性和局部化现象的新依据，更深入地深入一维KPZ的结构。在此过程中，将在概率、代数组合学、随机矩阵理论、可积系统、数论(自同构形式)之间建立新的联系。B.通过构造体现二维KPZ方程的对象(S)并提取其性质，通过离散系统的适当的尺度极限来进行二维KPZ方程的第一步。

It was proposed by Kardar, Parisi and Zhang in the 1980s that a large class of randomly growing interfaces exhibit universal fluctuations described mathematically by a nonlinear stochastic partial differential equation, which is now known as the Kardar-Parisi-Zhang or KPZ equation. Models within this class exhibit three basic mechanisms: growth as a function of the steepness of the interface, a smoothing effect modelled by Laplacian and local randomness modelled by white noise. Examples of physical systems which fall in this class are percolation of liquid in porous media, growth of bacteria colonies, currents in one dimensional traffic or liquid systems, liquid crystals etc.Remarkably the fluctuations of such random interfaces are governed by exponents and distributions that differ from the predictions given by the classical central limit theorem. In dimension one they are, surprisingly, linked to laws emerging from random matrix theory, as this was first exhibited by the work of Baik-Deift-Johansson, followed by a flurry of activity which set the framework of "determinantal processes". New exciting developments have taken place in the more recent years, making the first important steps into universality beyond determinantal models. In dimension two the situation is much less developed as governing exponents and distributions are not known and even the meaning of the two dimensional KPZ is not set in place.The goal of the project is twofold: A. To penetrate deeper into the structure of one dimensional KPZ via setting a robust framework to study fluctuations of non determinantal systems, attacking pending conjectures on multipoint correlations and exploring new grounds into the universality and localisation phenomena. In doing so, novel links between probability, algebraic combinatorics, random matrix theory, integrable systems, number theory (automorphic forms) will be made. B. To make the first steps in dimension two by constructing, via suitable scaling limits of discrete systems, the object(s) that incarnate the two dimensional KPZ equation and extract their properties.