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

该项目涉及随机生长界面的普遍大规模行为，这是描述真实物理过程的数学模型，如传播火锋、生长液晶、细菌菌落和咖啡污渍。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大颗粒系统，其中每个颗粒对应于一个分子或一个单独的细菌。我们测量的不精确和物理过程对许多因素的依赖通常由粒子的随机扰动来描述。当人们在足够长的一段时间后，从足够大的距离观察系统时，就可以观察到这种模型的大规模行为。在这种情况下，普遍性是指这样一个事实，即这种不断增长的界面表现出类似的大规模行为。对这一普遍行为的描述可以更好地理解自然界中真实的物理过程。最近概率论的突破，特别是在随机偏微分方程组和可积概率方面的突破，为研究这类数学模型提供了工具。当研究随机生长界面的极限行为时，Kardar-Parisi-Zhang(KPZ)普适性出现在非平衡统计力学中。在数学水平上，生长界面出现在定向随机聚合物的自由能、随机生长模型、相互作用粒子系统、随机Burgers方程和Hamilton-Jacobi-Bellman方程以及随机扰动的反应扩散方程中。根据模型的特点，可以猜想两个普适对象来控制这种界面：KPZ方程和KPZ不动点。随机偏微分方程组领域的最新发展使得证明了各种离散系统对奇异随机偏微分方程组的尺度极限。此外，可积概率方法(精确可解模型)给出了KPZ不动点的完整刻画。这个项目的目标是使用这些新的结果来研究这种随机增长界面的通用伸缩极限。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.