Selected Topics in Random Walk in Random Environment

TL;DR

This paper reviews the random walk in random environment (RWRE) model, focusing on multidimensional cases and recent advances in understanding conditions for ballistic behavior, highlighting its relevance across physics, biology, and engineering.

Contribution

It provides a comprehensive overview of the multidimensional RWRE model and discusses recent progress on ballisticity questions, a key open problem in the field.

Findings

Summarizes the state of the art in multidimensional RWRE

Highlights recent progress in ballisticity conditions

Discusses challenges due to loss of Markov property and non-reversibility

Abstract

Random walk in random environment (RWRE) is a fundamental model of statistical mechanics, describing the movement of a particle in a highly disordered and inhomogeneous medium as a random walk with random jump probabilities. It has been introduced in a series of papers by Chernov and Temkin as a model for DNA chain replication and crystal growth, and also as a model for turbulent behavior in fluids through a Lorentz gas description by Sinai. It is a simple but powerful model for a variety of complex large-scale disordered phenomena arising from fields such as physics, biology and engineering. While the one-dimensional model is well-understood, in the multidimensional setting, fundamental questions about the RWRE model have resisted repeated and persistent attempts to answer them. Two major complications in this context stem from the loss of the Markov property under the averaged measure as well as the fact that in dimensions larger than one, the RWRE is not reversible anymore. In these notes we present a general overview of the model, with an emphasis on the multidimensional setting and a more detailed description of recent progress around ballisticity questions.