Collaborative Research: Stochastic Interactions between Particles and Environments

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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.