Random Polymer Measures

这个项目是概率论和统计力学的严格方面的接口。该活动旨在研究具有复杂相互作用的系统的进化，例如在无序环境中移动的粒子，在交通中导航的汽车，生长晶体的表面或受感染组织的边界。复杂性是由模型中的随机性捕获的，无论是在粒子相互作用或晶体生长的环境中，还是在相互作用或生长过程本身。该项目的目的是发展控制这些系统的数学规律。有一个非常简单的例子，我们可以考虑在大量投掷硬币中正面的比例如何收敛于在一次投掷中得到正面的概率。（这一基本现象背后的数学规律被称为大数定律。）除了对概率论和一般数学的影响外，所提出的活动预计将对许多涉及随机或无序介质中随机运动的物理系统的理解产生直接影响。理解复杂的相互作用对科学和工程，从而对社会有着广泛的影响。本次活动的主题是随机介质中的随机运动。这一领域的模型已被数学家、自然科学家、社会科学家和工程师密切研究。它们的重要性源于与著名的kardar - paris - zhang （KPZ）方程和KPZ通用性类的严格数学联系。PI已经建立了能量-熵对偶性，并根据Busemann函数给出了这些变分公式的解。特别是对于一般权重分布的增长模型，PI证明了Busemann函数的几乎肯定存在性。PI将把这个存在性结果推广到正温度聚合物，随机环境中的随机行走，高维模型，以及随机热方程的连续设置。该奖项反映了美国国家科学基金会的法定使命，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。

This project is on the interface of probability theory and rigorous aspects of statistical mechanics. The proposed activity aims at investigating the evolution of systems with complex interactions, such as particles moving in a disordered environment, cars navigating their way through traffic, the surface of a growing crystal, or the boundary of an infected tissue. Complexity is captured by the randomness in the model, both in the environment in which the particles interact or the crystal grows, and in the interaction or growth process itself. The aim of the project is to develop the mathematical laws that govern such systems. To have a very simple example in mind, one can think of how the fraction of Heads in a large number of tosses of a coin will converge to the probability of getting Heads in one toss. (The mathematical law behind this basic phenomenon is known as the Law of Large Numbers.) Besides its impact on probability theory and mathematics in general, the proposed activity is expected to have a direct impact on the understanding of many physical systems involving random motion in random or disordered media. Understanding complex interactions has wide implications for science and engineering and thereby for society.The proposed activity is on the subject of random motion in random media. Models in this field have been intensely and concurrently studied by mathematicians, natural scientists, social scientists, and engineers. Their importance arises from rigorous mathematical connections to the celebrated Kardar-Parisi-Zhang (KPZ) equation, and the KPZ universality class. The PI has already established energy-entropy duality and produced solutions to these variational formulas in terms of Busemann functions. In particular, the PI proved almost sure existence of Busemann functions for the growth model with general weight distribution. The PI will work on generalizing this existence result to positive temperature polymers, random walk in random environment, higher dimensional models, and to the continuum setting of the stochastic heat equation.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.