RUI: First-passage Percolation and Other Disordered Systems

两种无序系统模型，第一通道渗透（FPP）和随机初始状态的随机伊辛动力学，是本建议的主要重点。关于基于晶格的FPP模型的几何特征的严格的无条件结果很难得到，这主要是由于与底层晶格的各向异性相关的技术困难。欧几里得FPP模型发生在齐次泊松过程构造的图上，在所有刚性运动下具有完全的统计不变性。从这个意义上说，欧几里得FPP是研究FPP几何性质的自然环境。本提案的目标之一是为欧几里得FPP改进与这些模型相关的关键波动指数的现有估计，最终目标是证明有趣的定性结果。另一个目标是利用欧几里得FPP的某种重标度关系来获得关于通过时间作为距离函数的随机单调性的结果。其他密切相关的开放问题正在研究中，比如泊松粒子构型的局部扰动如何影响最小化路径，以及“平行”测地线在合并之前通常走多远。在随机Ising模型中，动力学规则与随机初始状态相互作用，产生随时间演变的系统。任何特定时刻的状态都是与某些规则晶格的位置相关的+/-1自旋的构型；时刻0的构型使自旋为随机变量。对于这一建议的均匀铁磁模型，随着时间的推移，动力学倾向于使相邻位置的自旋一致性增加。研究的关键问题是系统发展到极限状态的速度和意义，以及配置的各种渗透特性作为时间的函数。无序系统是一类很大的概率模型，通常是由凝聚态物理和材料科学中出现的问题引起的。首通道渗流和其他密切相关的模型已经成为看似多种多样的现象的数学模型，如随机多孔介质（如含水层）的性质、癌性肿瘤的生长和脆性材料裂纹的扩展。Ising模型捕捉了磁化材料的基本特征。提出的工作旨在严格理解这些模型的一些基本数学性质。

Two models of disordered systems, first-passage percolation (FPP) and stochastic Ising dynamics with random initial state, are the main focus of this proposal. Rigorous unconditional results about the geometric features of lattice-based FPP models are hard to come by, largely due to technical difficulties associated with the anisotropy of the underlying lattice. Models of Euclidean FPP take place on graphs constructed from a homogeneous Poisson process and enjoy complete statistical invariance under all rigid motions. In this sense, Euclidean FPP is a natural setting in which to study geometric properties of FPP. One objective of this proposal is to refine, for Euclidean FPP, existing estimates of key fluctuation exponents associated with these models with the ultimate goal of proving interesting qualitative results. Another goal is to exploit a certain re-scaling relation that holds for Euclidean FPP to obtain results about the stochastic monotonicity of passage time viewed as a function of distance. Other closely related open questions under investigation concern issues such as how massively a local perturbation of the Poisson particle configuration affects minimizing paths and how far "parallel" geodesics typically travel before coalescence. In Stochastic Ising models, rules of dynamics interact with a random initial state to produce a system that evolves with time. The state at any particular time is a configuration of +/-1 spins associated with the sites of some regular lattice; the configuration at time 0 makes the spins i.i.d. random variables. For the homogeneous ferromagnetic models of this proposal, the dynamics tend to produce increasing agreement of spins for neighboring sites as time elapses. Key matters of investigation concern how quickly and in what sense the system evolves to a limiting state as well as various percolation properties of the configuration as a function of time. Disordered systems are a large class of probabilistic models generally motivated by problems arising in condensed matter physics and materials science. First-passage percolation and other closely related models have served as a mathematical model for phenomena as seemingly diverse as properties of randomly porous media (such as aquifers), the growth of cancerous tumors, and the propagation of cracks through brittle material. Ising models capture the basic features of magnetized materials. The proposed work aims at a rigorous understanding of some of the many fundamental mathematical properties of these models.