Collaborative Research: Stochastic Interactions between Particles and Environments

本文对随机过程的两个领域：随机环境中的随机游动和相互作用的粒子系统进行了研究。在前者中，粒子受其与非均匀随机介质的相互作用所驱动，而在后者中，粒子与存在的其他粒子相互作用。在一般的RWRE模型中，环境是晶格位置之间转移概率的集合，它是从某种混合的平移不变分布中选择的。另一方面，由第一个PI引入的一类沉积模型不仅为许多著名的粒子系统提供了一个统一的框架，而且对此类系统中出现的常见现象提供了更广泛的观点。在粒子系统中至关重要的一个物体，在精神上类似于随机环境中的随机游走者，即所谓的第二类粒子。这些领域的主要目标是理解环境中的随机性或由粒子-粒子相互作用引起的后果。人们可以考虑的基本问题包括0-1定律、大数定律、中心极限定理、大偏差等。这两个领域在精神和思想上非常接近，例如，不可逆性是这两个领域共同应用的主要障碍。有时，甚至可以通过同一系统的不同表示在两个领域之间建立直接联系。这项研究包含一个实例，其中表面生长过程被证明是RWRE模型的对偶，允许结果从后者流向前者。交互系统领域内的双重连接被认为是一个强大的工具。类似地，在相互作用的系统和RWRE之间进行比较，可以交换方法、见解和结果。除了概率直觉和技术往往与数学的其他领域密切相关外，这项研究对概率论、组合学、统计物理学和偏微分方程组理论也有直接影响。随机媒体和交互系统的领域也有各种工业、农业、社会学和生物应用。此外，这些领域的方法和问题往往很容易说明，而解决方案则需要复杂的工具。这使得这个主题吸引了研究生和年轻的研究人员，产生了更多对概率论的兴趣。

This research is conducted on two fields of stochastic processes: random walk in random environment (RWRE) and interacting particle systems. In the former, a particle is driven by its interaction with the non-homogeneous random medium, while in the latter, the particle interacts with other particles present. In the general RWRE model an environment is a collection of transition probabilities between lattice sites, and it is chosen from some mixing shift-invariant distribution. On the other hand, a class of deposition models, introduced by the first PI, not only provides a unified framework for many well-known particle systems, but also gives a broader view on common phenomena arising in such systems. An object of vital importance in particle systems, in spirit similar to a random walker in random environment, is the so-called second class particle. The main goal of these fields is the understanding of the consequences of the randomness in the environment, or caused by particle-particle interactions, respectively. Among the fundamental questions one can consider are the existence of 0-1 laws, law of large numbers, central limit theorems, large deviations, etc. The two fields are very close in spirit and ideas, just as an example, non-reversibility is a major obstacle in both fields to common applications. Sometimes even direct connections can be established between the two fields through different representations of the same system. This research contains an instance where a surface growth process is shown to be the dual of a RWRE model, allowing the flow of results from the latter to the former. Dual connections within the field of interacting systems are known to be a powerful tool. Analogously, drawing parallels between interacting systems and RWRE's allows for the exchange of methods, insights, and results.Besides the fact that probabilistic intuition and techniques are often of great relevance to other areas of mathematics, this research has a direct impact on probability theory, combinatorics, statistical physics, and the theory of partial differential equations. The fields of random media and interacting systems also have various industrial, agricultural, sociological, and biological applications. Moreover the methods and problems in these fields are often easy to state, while the solutions involve sophisticated tools. This makes the subject appealing to graduate students and young researchers, generating more interest in probability theory.