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。该项目将利用新工具来解决以前无法接触到的问题。该项目包括一系列更广泛的影响活动，包括组织科学、教育、多样性/公平/包容性和外联方案；为初级研究人员提供咨询和指导；以及在编辑和科学委员会任职。随机偏微分方程组、随机介质中的随机游动、相互作用的粒子系统、六顶点模型和Gibbs态是概率统计物理、平衡和非平衡统计物理、组合学、分析和表象理论的活跃研究领域。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。特别是，这个项目将探索(1)与边界接触的各种生长模型的不变测量和混合时间的性质，(2)多类粒子系统的行为和生长动力学中微扰的传播，以及(3)界面的波动，特别是随着各种应用而出现上、下偏差概率的可能性。通过将可积结构(例如杨-巴克斯特方程、对称函数、行列式过程、矩阵乘积ansasz)与概率方法(例如耦合、吉布斯性质、流体力学/随机PDE极限)相结合，PI将解决这两个领域以前仅靠任何一种方法无法解决的问题。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。

Probability, as a field, tries to address the question of how large complex random systems behave. An important class of probabilistic models deals 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 moves 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 new tools to solve previously inaccessible problems. The project includes a range of broader impact activities, including the organization of scientific, education, diversity/equity/inclusion, and outreach programs; advising and mentoring junior researchers; and serving on editorial and scientific boards and committees. Stochastic PDEs, random walks in random media, interacting particle systems, six vertex model, and Gibbs states are active areas of study within probability, equilibrium and non-equilibrium statistical 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 probe (1) the nature of invariant measures and mixing times for various growth models in contact with boundaries, (2) the behavior of multi-class particle systems and the propagation of perturbations in growth dynamics, and (3) the fluctuations of interfaces, in particular the likelihood of upper and lower deviation probabilities along with various applications. By marrying integrable structures (e.g. Yang-Baxter equation, symmetric functions, determinantal processes, matrix product ansastz) with probabilistic methods (e.g. couplings, Gibbsian properties, hydrodynamic / stochastic PDE limits) 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.