Structures and universalities around the Kardar-Parisi-Zhang equation

Kardar， Parisi和Zhang在20世纪80年代提出，一大类随机生长的界面表现出普遍波动，用非线性随机偏微分方程（现在称为Kardar-Parisi-Zhang或KPZ方程）在数学上描述。这类模型表现出三种基本机制：增长作为界面陡度的函数，由拉普拉斯模型模拟的平滑效应和由白噪声模型模拟的局部随机性。属于这一类的物理系统的例子有液体在多孔介质中的渗透、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，这些随机界面的波动是由指数和分布控制的，这些指数和分布与经典中心极限定理给出的预测不同。令人惊讶的是，在第一维中，它们与随机矩阵理论中出现的规律联系在一起，因为这首先是由Baik-Deift-Johansson的工作所展示的，随后是一系列活动，这些活动设置了“决定性过程”的框架。近年来出现了新的令人兴奋的发展，使决定模型向普遍性迈出了重要的第一步。在二维情况下，由于不知道控制指数和分布，甚至二维KPZ的含义也没有确定，情况就不那么发达了。该项目的目标是双重的：a.通过设置一个强大的框架来研究非确定性系统的波动，攻击关于多点相关性的未决猜想，并探索普遍性和局域化现象的新基础，从而深入了解一维KPZ的结构。在此过程中，概率论、代数组合学、随机矩阵理论、可积系统、数论（自同构形式）之间的新联系将被建立起来。B.通过离散系统的适当缩放限制，通过构建体现二维KPZ方程的对象并提取其属性，从而在二维中迈出第一步。

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.