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漫步。这个领域首先引起物理界的兴趣。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），均匀环境中扩散的经典结果（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。物理学家通过研究理想模型来解释这些行为，其中包括临时陷阱。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，我们发现了一些行为（收敛到稳定定律和老化），这些行为被认为是发生捕获的模型的普遍行为。如今，我们从Bouchaud陷阱模型中获得的知识使我们能够展示几种不同的随机环境（在Z和树上）中的随机行走模型，这些模型属于Bouchaud普适性类。***提出的研究计划的目标有两个：***-表明在随机环境中有更多的随机漫步模型属于Bouchaud普适性类。***-发现与减速发生的模型相关的新的限制行为。***对于程序的第一部分，一个关键方面是精确的捕获行为尚未被证明在Zd上随机环境中的随机漫步。这些模型是物理上最相关的模型（特别是在渗透簇上行走），从数学的角度来看，它们也是最具挑战性的。在Zd上找到任何一个新的随机环境下的随机行走模型，它属于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将是使用在随机环境中可逆随机行走的一些技术来研究不可逆行走的情况。***对于程序的第二部分，应该注意到有几个模型出现减速，但其极限行为不被认为是布绍普适性类的特征。一个关键的例子是Zd中关键渗透簇上的简单随机漫步。这可以说是随机环境中最自然和物理相关的随机行走模型。该模型受到了广泛的关注，但其极限行为却无法得到恰当的描述。该计划的长期目标之一是在高维上获得对这些行为的精确描述

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.**