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，火焰前锋的传播等，被认为是表现出各种普遍的性质，如果在某些特征的空间和时间尺度上观察。概率和统计物理学的大部分研究都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，缩放限制，相变，某些自然参数变化，收敛到平衡以及大偏差区域的行为。虽然主要重点是在概率论中发展新的想法，但一个关键目标是合并观点，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。该计划也有一个重要的教育组成部分，包括本科和研究生水平的课程开发，并指导研究生和博士后。该项目主要讨论三个主题。第一个主题包括随机增长模型，在局部粗糙化力的存在下表现出全局平滑机制，这些局部粗糙化力被认为表现出Kardar，Parisi和Zhang（KPZ）在一篇开创性论文中预测的某些普遍行为。PI将研究平面最后一次和第一次通过渗透的模型，该模型在平面晶格的顶点上放置随机权重，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普适类中的典型示例。 有一个爆炸性的活动，主要是围绕着这样的模型，这是可积的，承认某些显着的双射代数对象，如随机矩阵，杨图等。PI将追求几何的角度和概率工具，研究空间和时间的相关行为，这样的模型，以及如何在大偏差制度的最佳路径的几何变化。 第二个主题是关于自组织临界性的模型，其中系统在其自然演化下收敛到临界状态，而无需外部调整参数。继续以前的工作，PI将调查长期存在的关于无限晶格上相变的假设和有限版本的定量估计，随机沙堆模型和激活的随机行走，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相关多类型Laplacian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尔可夫链的逃逸率、谱行为和收敛到平衡点的指数的研究。考虑的例子包括在重力下扩散的粒子模型，在一个随机发展的潜力与连接到流体力学，随机游走的随机分形图，以及一类非单调自旋系统建模的'笼效应'在玻璃动力学，该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的学术价值和更广泛的影响审查标准。

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.