Scaling Limits of Growth in Random Media

概率作为一个领域，试图解决大型复杂随机系统如何表现的问题。一类重要的概率模型处理随机介质中的增长。这些可以用来模拟癌症如何在特定器官中生长，汽车如何在高速公路上的交通中移动，神经元如何在大脑中移动，或者疾病如何在人群中传播。该项目的目的是了解随机介质中增长的重要模型，包括与模型相关的统计分布，以及了解这些模型与什么样的系统相关。该项目将利用PI从许多数学领域开发的工具来解决以前无法解决的问题。随机偏微分方程，随机介质中的随机行走，相互作用粒子系统，六顶点模型和吉布斯状态是概率，平衡/非平衡统计物理，组合学，分析和表示论的活跃研究领域。这个项目涉及到这些领域的问题，并借鉴了这些领域的工具。特别是，本项目将（1）开发一种新的基于马尔可夫对偶的方法来证明微观模型的收敛性（2）证明KPZ方程及相关过程的尾部和大偏差界，并将其用于慢键问题等应用，（3）研究了随机环境中随机游动的标度行为，建立了FKPP方程与KPZ方程之间的关系，并研究了Gibbsian线系综的唯一性。通过结合可积概率和随机分析的方法，PI将解决这两个领域的问题，这两个领域是以前单独使用任何一种方法都无法解决的。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Probability, as a field, tries to address the question of how large complex random systems behave. An important class of probabilistic models deal with growth in random media. These can be used to model how a cancer grows in a particular organ, how a car moves through traffic on a highway, how neurons move through the brain, or how disease spreads through a population. The purpose of this project is to understand important models for growth in random media, both in terms of developing statistical distributions associated with the models, and in terms of understanding in what sort of systems these models are relevant. The project will leverage tools that the PI has been developing from a number of areas of mathematics to solve problems which were previously inaccessible.Stochastic partial differential equations, random walks in random media, interacting particle systems, six vertex model, and Gibbs states are active areas of study within probability, equilibrium / non-equilibrium statistics physics, combinatorics, analysis and representation theory. This project touches on problems in and draws upon tools from each of these areas. In particular, this project will (1) Develop a new Markov duality based method to prove convergence of microscopic models (including the six vertex model, dynamic ASEP and ASEP with inhomogeneous jump rates) to the KPZ equation, (2) Prove tail and large deviation bounds on the KPZ equation and related processes, and use these for applications like the slow bond problem, (3) Study scaling behavior for random walks in random environments and develop relationships between the FKPP and KPZ equations, as well as study the uniqueness of Gibbsian line ensembles. Through marrying methods from integrable probability and stochastic analysis, the PI will solve problems in both areas which were previously inaccessible from either approach alone.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.