Random Polymer Measures

这个项目的重点是概率论和统计力学的接口。该活动旨在研究具有复杂相互作用的系统的演化，例如在无序环境中移动的粒子，汽车在交通中导航，生长晶体的表面或受感染组织的边界。复杂性被模型中的随机性所捕获，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中。该项目的目的是开发管理此类系统的数学定律。举一个简单的例子，我们可以考虑在多次抛硬币中正面朝上的概率如何收敛到一次抛硬币中正面朝上的概率。除了对概率论和数学的影响外，该项目还将对理解许多涉及随机或无序介质中运动的物理系统产生直接影响。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。该项目为研究生和博士后提供了研究培训的机会。这项研究的主题是随机介质中的随机运动。在他最近的工作中，PI专注于研究无规聚合物模型，包括正温度和零温度。首要的主题是建立能量-熵对偶性，用Busemann函数求解这些变分公式，然后用这些解来描述大体积系统。PI已经在各种模型中构建了Busemann函数，并使用它们来分析无限测地线（或吉布斯测度，在正温度情况下），建立单力单解原理，并研究相关动力系统的稳定性和不稳定性。在这个项目中，PI将致力于将这些结果扩展到一个相当普遍的设置，将涵盖零和正温度，有向和无向，离散和连续，以及高维模型。PI还将开发使用Busemann函数来研究极限形状函数的规律性的方法，证明局部化，研究聚合物的大小和形状波动，并且，对于随机环境中的随机行走，描述马丁边界的结构。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进行评估，被认为值得支持的搜索.

This project is focused on the interface of probability theory and statistical mechanics. The 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 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. Besides its impact on probability theory and mathematics in general, the project will have a direct impact on the understanding of many physical systems involving motion in random or disordered media. Understanding complex interactions has wide implications for science and engineering and thereby for society. The project provides research training opportunities for graduate students and postdoctoral associates.This research is on the subject of random motion in random media. In his recent work, the PI has focused on the study of random polymer models, both in positive and zero temperature. The overarching theme was establishing energy-entropy duality, producing solutions to these variational formulas in terms of Busemann functions, then using these solutions to describe the large-volume systems. The PI already constructed the Busemann functions in a variety of models and used them to analyze infinite geodesics (or Gibbs measures, in the positive temperature case), establish one-force one-solution principles, and study the stability and instability of the related dynamical systems. In this project, the PI will work on extending these results to a quite general setting that will cover zero and positive temperature, directed and undirected, discrete and continuous, as well as higher-dimensional models. The PI will also develop methods to use Busemann functions to study the regularity of the limiting shape function, prove localization, study the size of the polymer and shape fluctuations, and, for random walk in random environment, describe the structure of the Martin boundary.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.