MATH 7780: Random Walks in Random Environments (Spring 2010)

Instructor: Jonathon Peterson

Random walks in random environments are a very simple model for the random motion of a particle in a random (non-homogeneous) medium. Despite the simple nature of the model and the similarity to the well studied model of classical random walks, random walks in random environments are surprisingly difficult. Moreover, random walks in random environments can exhibit behavior much different from that of a classical random walk such as

• Average drift in the opposite direction of transience
• Transient in one dimension but with zero speed
• Non-Gaussian limiting distributions

We will cover in this course some of the classical results for one-dimensional and multidimensional random walks in random environments as well as some of the more recent progress in the field. Depending on the interest of the class and time remaining we may also discuss other related models of random walks such as reinforced random walks and excited random walks.

In addition to learning the main results in random walks and random environments, we will learn some techniques that are useful in other fields as well, including:

• Explicit computations for reversible Markov chains
• Computations using one-step recursions
• Homogenization
• Large deviations
• Regeneration times (renewal theory)

Prerequisites for this course include a solid background in probability theory and a course in stochastic processes (in particular discrete time Markov chains).