RUI: First-passage Percolation and Other Disordered Systems

无序系统的两种模型，即首次通过渗流(FPP)和具有随机初始状态的随机伊辛动力学，是这一提议的主要焦点。基于晶格的FPP模型的几何特征很难得到严格的无条件结果，这在很大程度上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模型发生在由齐次泊松过程构成的图上，并且在所有刚性运动下都具有完全统计不变性。从这个意义上讲，欧几里得FPP是研究FPP几何性质的自然环境。这项提议的一个目标是完善欧几里德FPP对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所适用的某种重标度关系来获得关于作为距离的函数的通过时间的随机单调性的结果。其他正在调查的密切相关的悬而未决的问题涉及一些问题，例如泊松粒子构型的局部扰动对最小化路径的影响有多大，以及测地线在合并之前通常会走多远。在随机伊辛模型中，动力学规则与随机初始状态相互作用，产生一个随时间演化的系统。在任何特定时间的状态是与某些规则晶格的位置相关联的/-1自旋的组态；在时间0的组态使自旋I.I.D.随机变量。对于这一提议的均匀铁磁模型，随着时间的推移，动力学倾向于产生越来越多的相邻位置的自旋一致性。研究的关键问题涉及系统演化到极限状态的速度和在何种意义上，以及作为时间函数的配置的各种渗流特性。无序系统是一大类概率模型，通常是由凝聚态物理和材料科学中出现的问题引起的。首次通过渗流和其他密切相关的模型被用作各种现象的数学模型，如随机多孔介质(如含水层)的性质、癌症肿瘤的生长以及裂纹在脆性材料中的扩展。伊辛模型捕捉到了磁化材料的基本特征。这项拟议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.