Fractal Geometry for Dynamics on Random Media

当激光脉冲射入液晶时，从脉冲接触的地方产生一个不稳定区域，或者当墨渍在纸上扩散时，界面在局部无序环境中生长。统计力学严格理论的一个中心任务是分析物理系统的哪些特征是普遍的，这些特征为许多显然具有非常不同微观规格的模型所共有。在过去的四十年里，许多数学家和物理学家对kardar - paris - zhang普适类的理解已经有了很大的进步，它涉及到许多具有局部随机性的表面。这些进步通常是借助可积技术的使用，其中有时通过代数工具或精确公式分析相当特殊的模型。首席研究员将分析某些重要的普适类，包括kardar - paris - zhang类，使用概率证明方法与有限但关键的可积输入相一致。在随机介质上动力学的尺度限制中产生的某些随机分形几何的结构将由此得到解释。提出的分形几何的概率研究有三个主要组成部分。第一种是关于无序随机介质中的异常输运。也就是说，当一个粒子在一个随机的、高度无序的环境中，以一致的趋势向一个首选方向移动时，捕获是一个重要的现象，它中断甚至有时消除了线性进程。这一现象将通过对至少2维整数晶格中超临界渗流无限团簇上的偏置运动的详细研究加以阐明。第二个方向涉及缩放的最后通道渗透。通过使用概率证明模式（如随机能量剖面的重采样和路径上的外科手术）研究穿过随机场的极端路径的缩放随机几何，将探索卡尔达-帕里西-张普适的丰富方面。最后的主成分将处理噪声敏感性和最后通道渗透。研究随机环境噪声扰动下最后通道渗流路径的稳定性和敏感性，加强两个关键理论：kardar - paris - zhang通用性和离散谐波分析之间的桥梁。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

When a laser pulse is fired into a liquid crystal, causing an unstable region to grow from where the pulse makes contact, or when the stain of an ink blot spreads on paper, interfaces grow in the presence of a locally disordered environment. A central task of the rigorous theory of statistical mechanics is to analyze which features of physical systems are universal, shared by many models with apparently very different microscopic specifications. Understanding of the Kardar-Parisi-Zhang universality class, which concerns many surfaces growing with local randomness, has been greatly advanced by many mathematicians and physicists over the last forty years. Often these advances have been aided by the use of integrable techniques, in which sometimes rather special models are analyzed via algebraic tools or exact formulas. The Principal Investigator will analyze certain important universality classes, including the Kardar-Parisi-Zhang class, by using probabilistic proof methods in unison with limited but crucial integrable inputs. The structure of certain random fractal geometries arising in scaling limits of dynamics on random media will thus be explicated. The proposed probabilistic investigation of fractal geometries has three principal components. The first concerns anomalous transport in disordered random media. Namely, when a particle with a uniform tendency to move in a preferred direction does so in a milieu that is random and strongly disordered, trapping is a vital phenomenon that interrupts and sometimes eliminates linear progress. The phenomenon will be elucidated by a detailed inquiry into biased motion on the infinite cluster of supercritical percolation in the integer lattice of dimensions at least two. The second direction concerns scaled last passage percolation. Rich aspects of Kardar-Parisi-Zhang universality will be explored by studying the scaled random geometry of extremal paths moving through random fields using probabilistic modes of proof such as resampling of random energy profiles and surgery on paths. The final principal component will address noise sensitivity and last passage percolation. The stability and sensitivity of last passage percolation paths under perturbation by noise of the random environment will be studied, strengthening the bridge between two key theories: Kardar-Parisi-Zhang universality and discrete harmonic analysis.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.