Selected Topics in randomwalks in random environment

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 randomwalk with random jump probabilities. It has been introduced in a series of papers as a model of DNA chain replication and crystal growth (see Chernov [10] and Temkin [51, 52]), and also as a model of turbulent behavior in fluids through a Lorentz gas description (Sinai 1982 [42]). 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.