The Kardar-Parisi-Zhang (KPZ) Universality of Random Growing Interfaces

该项目关注随机生长界面的普遍大规模行为，这些界面是描述真实的物理过程的数学模型，如传播的火锋，生长的液晶，细菌菌落和咖啡渍。这些过程通常使用统计力学方法建模，即通过考虑相互作用的粒子的大系统，其中每个粒子对应于分子或单个细菌。我们测量的不精确性和物理过程对许多因素的依赖性通常由粒子的随机扰动来描述。当人们在足够长的时间段之后从足够大的距离观察系统时，可以观察到这种模型的大尺度行为。在这种情况下，普遍性是指这样一个事实，即这种不断增长的接口表现出类似的大规模行为。对这种普遍行为的描述使我们更好地理解自然界中真实的物理过程。近年来概率论的突破，特别是在随机偏微分方程和可积概率方面的突破，为研究这类数学模型提供了工具。Kardar-Parisi-Zhang（KPZ）普适性在非平衡统计力学中研究随机增长界面的极限行为时产生。在数学层面上，生长界面出现在定向无规聚合物的自由能、无规生长模型、相互作用粒子系统、随机Burgers和Hamilton-Jacobi-Bellman方程以及随机扰动反应扩散方程中。根据模型的特点，构造了两个统一的对象：KPZ方程和KPZ不动点。随机偏微分方程领域的最新发展允许证明各种离散系统的尺度极限奇异随机偏微分方程。此外，可积概率方法（精确可解模型）提供了KPZ不动点的完整刻画。这个项目的目标是研究通用的缩放限制，这种随机增长的接口使用这些新的result.This奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

The project concerns universal large-scale behavior of random growing interfaces, which are mathematical models that describe real physical processes such as propagating fire fronts, growing liquid crystals, bacterial colonies and coffee stains. These processes are typically modeled using the statistical mechanics approach, i.e. by considering large systems of interacting particles, where each particle corresponds to a molecule or an individual bacterium. Imprecision of our measurements and dependence of physical processes on many factors are typically described by random perturbations of particles. Large-scale behavior of such models is observed when one looks at the systems from a sufficiently large distance, after a sufficiently long period of time. Universality in this case refers to the fact that such growing interfaces exhibit similar large-scale behavior. Description of this universal behavior gives a better understanding of the real physical processes in nature. Recent breakthroughs in probability theory, particularly in stochastic partial differential equations and integrable probability, have provided the tools to study such mathematical models.The Kardar-Parisi-Zhang (KPZ) universality arises in non-equilibrium statistical mechanics, when studying limiting behavior of random growing interfaces. On the mathematical level, growing interfaces appear in free energies of directed random polymers, random growth models, interacting particle systems, stochastic Burgers and Hamilton-Jacobi-Bellman equations, and stochastically perturbed reaction-diffusion equations. Depending on the characteristics of models, two universal objects are conjectured to govern such interfaces: the KPZ equation and the KPZ fixed point. Recent developments in the area of stochastic PDEs allow to prove scaling limits of various discrete systems to singular stochastic PDEs. Moreover, methods of integrable probability (exactly solvable models) provided a complete characterization of the KPZ fixed point. The goal of this project is to study universal scaling limits of such random growing interfaces using these new results.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.