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Scaling limits of growth in random media

扩大随机介质的增长极限

基本信息

批准号：
2246576
负责人：
Ivan Corwin
金额：
$ 50万
依托单位：
Columbia University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2023
资助国家：
美国
起止时间：
2023-07-01 至 2028-06-30
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2246576&HistoricalAwards=false
关键词：
Scaling limits growth random media

项目摘要

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.

概率作为一个领域，试图解决大型复杂随机系统的行为问题。一类重要的概率模型涉及随机介质的增长。这些可用于模拟癌症如何在特定器官中生长、汽车如何在高速公路上行驶、神经元如何在大脑中移动，或者疾病如何在人群中传播。该项目的目的是了解随机介质增长的重要模型，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与何种类型的系统相关方面。该项目将利用新工具来解决以前无法解决的问题。该项目包括一系列影响更广泛的活动，包括组织科学、教育、多样性/公平/包容性和外展计划；为初级研究人员提供建议和指导；并在编辑和科学委员会中任职。随机偏微分方程、随机介质中的随机游走、相互作用粒子系统、六顶点模型和吉布斯态是概率、平衡和非平衡统计物理、组合学、分析和表示理论中的活跃研究领域。该项目涉及这些领域中的问题并利用了这些领域的工具。特别是，该项目将探讨（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扰动的传播，以及（3）界面的波动，特别是上偏差概率和下偏差概率以及各种应用的可能性。通过将可积结构（例如 Yang-Baxter 方程、对称函数、行列式过程、矩阵乘积 ansastz）与概率方法（例如耦合、吉布斯性质、流体动力学/随机 PDE 极限）相结合，PI 将解决这两个领域中的问题，而这些问题以前仅通过任何一种方法都无法解决。该奖项反映了 NSF 的法定使命，并已被通过使用基金会的智力优点和更广泛的影响审查标准进行评估，认为值得支持。