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的行为方式问题。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的器官中生长，汽车如何在高速公路上的车流中行驶，神经元如何在大脑中移动，或者疾病如何在人群中传播。本项目的目的是了解随机介质中增长的重要模型，无论是在开发与模型相关的统计分布方面，还是在理解这些模型在哪种系统中相关方面。该项目将利用PI从多个数学领域发展而来的工具来解决以前无法解决的问题。随机偏微分方程、随机介质中的随机游动、相互作用的粒子系统、六个顶点模型和吉布斯状态是概率论、平衡/非平衡统计物理、组合学、分析和表示理论中的活跃研究领域。这个项目涉及到这些领域中的问题，并利用了这些领域的工具。具体地说，这个项目将(1)发展一种新的基于马尔可夫对偶的方法来证明微观模型(包括六个顶点模型、动态ASEP和具有非均匀跳跃速率的ASEP)到KPZ方程的收敛；(2)证明KPZ方程和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)研究随机环境中随机游动的标度行为，建立FKPP和KPZ方程之间的关系，以及研究吉布斯线系综的唯一性。通过结合可积概率和随机分析的方法，PI将解决这两个领域以前仅靠这两种方法无法解决的问题。这一奖项反映了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。

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.