Large Scale Asymptotics of Random Spatial Processes: Scaling Exponents, Limit Shapes, and Phase Transitions

许多自然过程，例如细菌的生长、多孔介质中的流体扩散、随机介质中的定向聚合物、火焰锋的传播等，如果在某些特征空间和时间尺度上观察，被认为表现出各种普遍特性。概率和统计物理学的大部分研究涉及研究配备空间几何的随机结构，期望对上述和其他一些自然现象进行建模。该项目概述的研究计划旨在研究围绕此类随机空间模型各个方面的广泛问题，包括相关结构、尺度限制、某些自然参数变化时的相变、收敛到平衡以及大偏差状态下的行为。虽然主要重点是发展概率论中的新颖思想，但一个关键目标是融合观点并在数学、统计物理和理论计算机科学的各个领域之间建立新的桥梁。该计划还具有重要的教育组成部分，包括本科生和研究生水平的课程开发以及指导研究生和博士后。该项目广泛讨论三个主题。第一个主题包括随机增长模型，该模型在存在局部粗糙力的情况下表现出全局平滑机制，据信表现出卡达尔、帕里西和张（KPZ）在一篇开创性论文中预测的某些普遍行为。 PI 将研究平面最后和第一通道渗透模型，该模型对平面晶格的顶点施加随机权重，并考虑分别产生最大或最小能量的顶点之间的路径，并且被认为是 KPZ 普适性类中的典型示例。 活动呈爆炸式增长，主要围绕此类模型的少数示例，这些模型是可积的，承认代数对象（例如随机矩阵、杨图等）的某些显着双射。 PI 将追求几何视角并开发概率工具来研究此类模型的空间和时间相关行为，以及最佳路径的几何形状在大偏差情况下如何变化。 第二个主题涉及自组织临界模型，其中自然演化下的系统无需外部调整参数即可收敛到临界状态。继续之前的工作，PI 将研究关于无限晶格相变的长期猜想以及有限版本的定量估计，针对随机沙堆模型和激活随机游走，这是自组织临界性的两个范式示例。还提出了相关多类型拉普拉斯增长模型的演化自相似界面的研究，其中增长率由随机游走的调和测度控制。最后一个主题是关于随机游走和有限马尔可夫链的逃逸率、谱行为和收敛到平衡相关的指数的研究。考虑的例子包括在重力作用下以随机演化势扩散的粒子模型，与流体力学、随机分形图上的随机游动以及一类模拟玻璃动力学中“笼效应”的非单调自旋系统，与矩阵上的随机游走和定向渗透有关。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

Many natural processes such as growth of bacteria, fluid spreading in a porous medium, directed polymers in random media, propagation of flame fronts and so on, are believed to exhibit various universal properties if observed at certain characteristic spatial and time scales. Much of the research in probability and statistical physics involves investigating random structures equipped with spatial geometry, expected to model some natural phenomena as above and others. The research program outlined in the project aims to study a wide range of problems around various aspects of such random spatial models including correlation structure, scaling limits, phase transitions as certain natural parameters are varied, convergence to equilibrium as well as behavior in the large deviation regimes. While the main focus is on developing novel ideas in probability theory, a key goal is to merge perspectives and develop new bridges between various areas of mathematics, statistical physics and theoretical computer science. The program also has a significant education component including curriculum development at undergraduate and graduate levels, and mentoring graduate students and postdocs. The project broadly discusses three topics. The first theme includes models of random growth exhibiting a global smoothing mechanism in presence of local roughening forces believed to exhibit certain universal behavior predicted in a seminal paper by Kardar, Parisi and Zhang (KPZ). The PI will study models of planar last and first passage percolation, which puts random weights on the vertices of a planar lattice and considers paths between vertices which accrue maximum or minimum energies respectively, and are believed to be canonical examples in the KPZ universality class. There has been an explosion of activity, mostly around a handful of examples of such models, which are integrable, admitting certain remarkable bijections to algebraic objects such as random matrices, Young diagrams and so on. The PI will pursue a geometric perspective and develop probabilistic tools to study spatial and temporal correlation behavior for such models as well as how the geometry of optimal paths change in large deviation regimes. The second theme concerns models of self organized criticality where systems under their natural evolution converge to a critical state without external tuning of parameters. Continuing previous work, the PI will investigate long standing conjectures about phase transitions on infinite lattices and quantitative estimates for finite versions, for the stochastic sandpile model and activated random walk, two paradigm examples of self-organized criticality. The study of evolving self-similar interfaces of related multi-type Laplacian growth models where growth rate is governed by harmonic measure of random walk is also proposed. The final topic is about the study of exponents related to rate of escape, spectral behavior and convergence to equilibrium for random walks and finite Markov chains. Examples considered include models of particles diffusing under gravity in a random evolving potential with connections to fluid mechanics, random walks on random fractal graphs as well as a class of non-monotone spin systems modeling the 'cage effect' in glassy dynamics, with connections to random walk on matrices and oriented percolation.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.