Collaborative Research: Stochastic Interactions between Particles and Environments

本文对随机过程的两个领域进行了研究：随机环境中的随机行走（RWRE）和相互作用粒子系统。在前者中，粒子由其与非均匀随机介质的相互作用驱动，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，环境是晶格点之间转移概率的集合，它是从一些混合移位不变分布中选择的。另一方面，由第一次PI引入的一类沉积模型不仅为许多已知的粒子系统提供了统一的框架，而且为这些系统中出现的常见现象提供了更广阔的视角。在粒子系统中有一个极其重要的对象，其精神类似于随机环境中的随机步行者，即所谓的第二类粒子。这些领域的主要目标是理解环境随机性的后果，或者分别由粒子-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定律的存在性、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这两个领域之间建立直接联系。本研究包含一个实例，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。众所周知，相互作用系统领域中的双重连接是一种强大的工具。类似地，在交互系统和RWRE之间绘制相似之处允许交换方法、见解和结果。除了概率直觉和技术经常与其他数学领域密切相关的事实外，这项研究对概率论、组合学、统计物理和偏微分方程理论有直接影响。随机介质和相互作用系统领域也有各种工业、农业、社会学和生物学应用。此外，这些领域的方法和问题往往很容易表述，而解决方案涉及复杂的工具。这使得这门学科吸引了研究生和年轻的研究人员，对概率论产生了更大的兴趣。

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.