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

许多自然过程，如细菌的生长、流体在多孔介质中的传播、定向聚合物在随机介质中的传播、火焰前锋的传播等，如果在特定的空间和时间尺度上进行观察，被认为具有各种普遍的性质。概率和统计物理学的大部分研究涉及研究配备了空间几何的随机结构，预计这些结构将模拟上述和其他一些自然现象。该项目概述的研究计划旨在研究此类随机空间模型的各个方面的广泛问题，包括相关结构、比例限制、某些自然参数变化时的相变、收敛到平衡以及在大偏差区域中的行为。虽然主要的重点是在概率论中发展新的想法，但一个关键的目标是融合观点，并在数学、统计物理和理论计算机科学的不同领域之间建立新的桥梁。该计划还包括一个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。该项目大致讨论了三个主题。第一个主题包括随机增长模型，该模型展示了在存在局部粗化力的情况下的全局平滑机制，这些粗化力被认为展示了Kardar，Parisi和Zhang(KPZ)的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次通过渗流模型，该模型对平面晶格的顶点施加随机权重，并考虑顶点之间分别积累最大或最小能量的路径，被认为是KPZ普适性类中的典型例子。最近出现了一场爆炸式的活动，主要是围绕少数几个此类模型的例子，这些模型是可积的，允许某些显著的双射到代数对象，如随机矩阵、杨图等。PI将追求几何观点，并开发概率工具来研究此类模型的空间和时间相关行为，以及最优路径的几何形状在大偏差区域中如何变化。第二个主题涉及自组织临界性模型，在这种模型中，系统在其自然演化下收敛到临界状态，而无需外部参数调整。在之前工作的基础上，PI将调查关于无限晶格上相变的长期猜测和有限版本的定量估计，对于随机沙堆模型和激活的随机行走，这是自组织临界性的两个范例。研究了增长率由随机游动的调和测度控制的多类拉普拉斯增长模型的演化自相似界面。最后一个主题是关于随机游动和有限马氏链的逃逸率、谱行为和收敛到平衡的指数的研究。被考虑的例子包括粒子在重力下以随机演化势扩散的模型，与流体力学有关，随机分形图上的随机游动，以及一类非单调自旋系统，模拟玻璃动力学中的“笼子效应”，与矩阵上的随机游动和定向渗流有关。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.