Random Walks in Random Environments

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 beautiful area of RWRE. The principal model to be discussed is a random walk with nearest-neighbor jumps in independent identically distributed (i.i.d.) random environment in one dimension, although we shall also comment on some extensions and generalizations. The focus is on rigorous results; however, heuristics is used freely to motivate the ideas and explain the approaches and proofs. In a few cases, sketches of the proofs have been included, which should help the reader to appreciate the flavor of results and methods.