Random walks in random environments and traps

所提出的研究计划涉及随机环境中的随机游动。 这个领域首先受到物理学界的关注。由于自然界中遇到的大多数材料都是无序的（多孔岩石，凝胶，导体），因此在均匀环境中扩散的经典结果（例如收敛到布朗运动）在许多物理相关的例子中是无效的。 这些异常行为是由于环境中自然出现的陷阱的存在。物理学家通过研究包含陷阱的理想化模型来解释这些行为。这些模型后来被称为布绍陷阱模型，也被数学家广泛研究。 通过对这些模型的研究，我们发现了几种行为（收敛到稳定律和老化），这些行为被认为是普遍存在于发生捕获的模型中的。如今，我们已经从布绍陷阱模型获得的知识，使我们能够展示几种不同的模型，随机环境中的随机游动（Z和树）属于布绍普适类。 拟议的研究计划的目标是双重的： - 证明随机环境中的随机游动的更多模型属于Bouchaud普适类。 - 查找与发生减速的模型相关的新限制行为。 对于程序的第一部分，一个关键的方面是Zd上随机环境中随机游动的精确捕获行为尚未得到证明。这些模型是物理上最相关的模型（特别是在渗流簇上的行走），从数学的角度来看，它们也是最具挑战性的。在Zd上的随机环境中找到任何新的随机游动模型都属于Bouchaud普适类，这将是一个非常重要的结果。在这个方向上的另一个重要发展是使用一些为随机环境中的可逆随机游动开发的技术来研究不可逆游动的情况。 对于该程序的第二部分，应该注意的是，有几个模型会发生减速，但限制行为并不被认为是Bouchaud普适性类的特征。一个关键的例子是在Zd中临界渗流簇上的简单随机游走。这可以说是随机环境中最自然和物理相关的随机游走模型。这个模型已经受到了很多关注，但限制行为不能被正确地描述。该计划的长期目标之一是获得这些行为的高维精确描述。

The proposed research program concerns itself with random walks in random environments. This field first received interest from the physics community. Since most materials encountered in nature are disordered (porous rocks, gels, conductors) the classical results from diffusions in homogeneous environments (such as convergence to a Brownian motion) are not valid in many physically relevant examples. These anomalous behaviors are due to the existence of traps naturally appearing in the environment. These behaviors were explained by physicists by studying idealized models where traps are included ad-hoc. These models came to be known as Bouchaud trap models and were also studied extensively by mathematicians. Through the study of these models, several behaviors were exhibited (convergence to stable laws and aging) which are believed to be universal for models where trapping occurs. Nowadays, the knowledge we have obtained from the Bouchaud trap models has allowed us to exhibit several different models of random walks in random environments (on Z and trees) belonging to the Bouchaud universality class. The goal of the proposed research program is two-fold: -Show that more models of random walks in random environments belong to the Bouchaud universality class. -Find new limiting behaviors associated with models in which a slowdown occurs. For the first part of the program, one critical aspect is that precise trapping behaviors have not yet been proved for random walks in random environments on Zd. These models are the most physically relevant ones (especially walks on percolation clusters), and they are also the most challenging from a mathematical perspective. Finding any new model of random walk in random environment on Zd belongs to the Bouchaud universality class would be a very important result. Another important development in this direction would be to use some of the techniques developed for reversible random walks in random environments to study the case of non-reversible walks.  For the second part of the program, it should be noted that there are several models in which slowdowns occur but where the limiting behaviors are not believed to be characteristic of the Bouchaud universality class. One key example is the simple random walk on the critical percolation cluster in Zd. This is arguably the most natural and physically relevant model of random walk in random environment. This model has received a lot of attention and yet the limiting behavior cannot not be described properly. One of the longer-term goals of this program is to obtain a precise description of these behaviors in high dimensions.