Random Polymer Measures

这项建议是关于概率论和统计力学的严格方面的接口。这项拟议的活动旨在研究具有复杂相互作用的系统的进化，例如粒子在无序环境中移动，汽车在交通中导航，生长的晶体的表面，或受感染组织的边界。复杂性被模型中的随机性所捕捉，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这类系统的数学定律。头脑中有一个非常简单的例子，你可以想一想，在一次抛硬币中，人头的比例如何会聚为一次抛出人头的可能性。(这一基本现象背后的数学定律称为大数定律。)除了对概率论和一般数学的影响外，拟议的活动预计将对许多物理系统的理解产生直接影响，这些系统涉及随机或无序介质中的随机运动。了解复杂的相互作用对科学和工程有广泛的影响，因此对社会也有影响。建议的活动是关于随机介质中的随机运动的主题。数学家、自然科学家、社会科学家和工程师对这一领域的模型进行了深入而同时的研究。它们的重要性源于与著名的Kardar-Parisi-Zhang(KPZ)方程和KPZ普适性类的严格数学联系。PI已经建立了能量-熵对偶，并用Busemann函数给出了这些变分公式的解。特别地，PI证明了具有一般权分布的增长模型的Busemann函数的存在性。PI下一步将把这个存在结果推广到正温度聚合物，以及更高维的模型。结果，KPZ波动指数终于可以用于一般模型，允许建立模型的一些普遍性质，超越了狭隘的显式可解情况集。这也为在随机热方程的连续介质中建立Busemann函数的存在性打开了大门，使得人们能够推导随机Burgers方程解的整体存在性，以及直接获得KPZ涨落结果，而不是通过离散模型的近似方案。

This proposal 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 next generalize this existence result to positive temperature polymers, as well as higher dimensional models. As a result, KPZ fluctuation exponents become at last accessible for general models, allowing to establish some universal properties for models beyond the narrow set of explicitly solvable cases. This also opens the door to establishing existence of Busemann functions in the continuum setting of the stochastic heat equation, enabling one to deduce global existence of a solution to the stochastic Burgers equation, as well as access KPZ fluctuation results directly rather than through approximation schemes via discrete models.