RUI: First-passage Percolation and Other Disordered Systems

两种模型的无序系统，首次通过渗流（FPP）和随机伊辛动力学与随机初始状态，是这个建议的主要焦点。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，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模型发生在由齐次泊松过程构造的图上，并且在所有刚性运动下具有完全的统计不变性。在这个意义上，欧几里德FPP是一个自然的设置，其中研究的几何性质的FPP。这个建议的一个目标是完善，欧几里德FPP，现有的估计与这些模型相关的关键波动指数证明有趣的定性结果的最终目标。另一个目标是利用一定的重新缩放关系，持有欧几里德FPP获得的结果的随机单调性的通过时间视为距离的函数。其他密切相关的开放式问题正在调查关注的问题，如如何大规模的局部扰动的泊松粒子配置影响最小化路径和多远的“平行”测地线通常走在合并之前。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个规则晶格的位置相关联的+/-1自旋的配置;在时间0的配置使自旋i.i.d.随机变量对于该提议的均匀铁磁模型，随着时间的推移，动力学倾向于产生相邻位置的自旋的增加的一致性。调查的关键问题关注如何快速和在什么意义上的系统演变到一个极限状态，以及各种渗流性能的配置作为时间的函数。 无序系统是一大类概率模型，通常由凝聚态物理和材料科学中出现的问题所激发。首次通过渗流和其他密切相关的模型已经作为一个数学模型的现象，似乎不同的随机多孔介质（如含水层）的属性，癌性肿瘤的生长，并通过脆性材料的裂纹传播。伊辛模型捕捉磁化材料的基本特征。拟议的工作旨在严格理解这些模型的许多基本数学性质中的一些。

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.