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Probability seminar: Random Walks in a Random Environment (BM)



The probability seminar aims at exposing students to advanced probability theory and selected applications. One topic will be selected each year. The topic for the academic year 2017-2018 is Random Walks in Random Environments (RWRE).

RWRE are random walks on lattices with random transition probabilities, models for motion of particles through highly irregular media due to defects, impurities or fluctuations. Fundamental examples of Markov processes with random transitions, they were introduced in the 1970s motivated by some problems in biology, crystallography and metal physics, but later applications have spread through numerous areas. They represents a very active and challenging research area in probability and physics of disordered systems.

The focus of this course is on key well-understood models and related techniques which can be useful in other fields. The principal model to be discussed is a random walk with nearest-neighbor jumps in independent identically distributed random environment on the one dimension lattice. Despite being the simplest model it shows a surprisingly reach behavior. Extensions and generalizations will be touched and a view of the state of the art in this area will be given.


Familiarity with basic concepts in probability theory and basics of Markov chains and random walks.


The course material is based on the following key references together with selected topical research papers
which will be given during the course:

  • O. Zeitouni, Random Walks in Random Environment, Lectures on Probability Theory and Statistics,
    Ecole dEte' de Probabilite's de Saint-Flour XXXI - 2001, Lecture Notes in Mathematics, Volume 1837,
    Part II, (2004).
    Available version at the author’s webpage

  • J. Peterson, Lecture Notes on Random Walks in Random Environments, (2013). Available at the
    webpage of the author: webpage.

  • F. den Hollander, Large Deviations, Fields Institute Monographs, vol. 14.S, (2000).

Plan and exam

Four introductory lectures plus another one at an advanced stage of the course are given by Avena and
den Hollander.

All the other lectures are given by the participants. Each participant gives 2 presentations:

  • one dedicated to some aspects of the general theory,

  • one chosen from the list of topics presented during the course.

For the second presentation, a short hand-out (to be distributed to the other participants) must be prepared
as well.

The final grade is based on active participation, the two presentations and the hand-out.