  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.
