
TL;DR
This paper introduces the study of Random Walks in Random Environments (RWRE), a model for transport processes in irregular media, highlighting key results, methods, and extensions in this active research area.
Contribution
It provides a comprehensive overview of RWRE, focusing on rigorous results, methods, and recent developments in one-dimensional models with i.i.d. environments.
Findings
Rigorous results on RWRE behavior
Heuristic explanations and proof sketches
Extensions to higher dimensions and generalizations
Abstract
Random walks provide a simple conventional model to describe various transport processes, for example propagation of heat or diffusion of matter through a medium. However, in many practical cases the medium is highly irregular due to defects, impurities, fluctuations etc., and it is natural to model this as random environment. In the random walks context, such models are referred to as Random Walks in Random Environments (RWRE). This is a relatively new chapter in applied probability and physics of disordered systems, initiated in the 1970s. Early interest was motivated by some problems in biology, crystallography and metal physics, but later applications have spread through numerous areas. After 30 years of extensive work, RWRE remain a very active area of research, which has already led to many surprising discoveries. The goal of this article is to give a brief introduction to the…
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