KPZ Universality

一维KPZ普适类包含随机生长模型、定向随机聚合物自由能、随机Hamilton-Jacobi-Bellman方程、随机Burgers方程、随机摄动反应扩散方程和相互作用粒子模型。在物理层面上，KPZ生长出现在森林火灾前线、细菌菌落边界、液晶和咖啡环等广泛的现象中。该类的特征是动态标度指数z=3/2。大约15年前的一些突破导致了一些模型波动的精确分布，并对整个类进行了推测外推。令人惊讶的是，这些分布原来是最近在随机矩阵理论中发现的，现在已经在物理实验中观察到。6年前，有了第二组突破，其中几个可调节不对称的模型得到了部分解决，导致KPZ方程本身的各种初始条件的精确分布，KPZ方程是一个非线性随机偏微分方程，在80年代中期作为经典连续体模型引入课堂。同时，在KPZ方程本身的适定性上的突破也为他赢得了2014年菲尔兹奖。在数学界和物理界都有激烈的活动，但这是一个不寻常的领域，数学家经常能够在物理问题上领先于物理学家。******第三次KPZ革命现在开始了，因为我们的团队终于能够访问所有精确公式背后的不变马尔可夫过程，KPZ不动点。在过去的几十年里，概率和统计物理的进展一直被这些可积不动点所主导，如SLE、布朗图和正弦核过程，它们为大波动类提供了解释。KPZ不动点承诺为KPZ通用性类做同样的事情。******本提案的目标是发展KPZ不动点的一般精确公式，深入了解波动的普适性，扩展KPZ方程本身的弱普适性，研究离散模型在欧拉尺度上从Burgers方程的熵解到这些新解的交叉，并开始访问更高维度的问题。

The one dimensional KPZ universality class contains random growth models, directed random polymer free energies, stochastic Hamilton-Jacobi-Bellman equations, stochastic Burgers' equations, stochastically perturbed reaction-diffusion equations, and interacting particle models. At the physical level, KPZ growth appears in phenomena as wide ranging at forest fire fronts, bacterial colony boundaries, liquid crystals, and coffee rings. The class is characterized by the unusual dynamic scaling exponent z=3/2. A number of breakthroughs about 15 years ago led to a few exact distributions of fluctuations for a few models, with conjectural extrapolation to the whole class. The distributions, surprisingly, turned out to be those recently discovered in random matrix theory, and have now been observed in physical experiments. 6 years ago there was a second group of breakthroughs in which several models with adjustable asymmetry were partially solved leading to exact distributions for various initial conditions for the KPZ equation itself, a non-linear stochastic partial differential equation introduced in the mid 80's as a canonical continuum model in the class. Concurrent breakthroughs on the well-posedness of the KPZ equation itself led to a 2014 Fields medal. There has been intense activity both in the mathematics and physics communities, but this is an unusual area where mathematicians have often been able to take the lead from physicists, on physical problems. ******A third KPZ revolution is now beginning, as our group has finally been able to access the invariant Markov process behind all the exact formulas, the KPZ fixed point. In the last few decades, progress in probability and statistical physics has been dominated by such integrable fixed points, such as SLE, the Brownian map, and the sine kernel process, which provide explanations for large fluctuation classes. The KPZ fixed point promises to do the same for the KPZ universality class. ******The goal of this proposal is to develop the general exact formulas for the KPZ fixed point, to gain insight into the universality of the fluctuations, to extend the weak universality of the KPZ equation itself, to study the crossover of discrete models from entropy solutions of Burgers' equation at the Euler scale, to these new solutions, and to begin to access problems in higher dimensions.