Random walks in random environment. (English) Zbl 1060.60103

Zeitouni, Ofer

Picard, Jean (ed.), Lectures on probability theory and statistics. Ecole d’Eté de probabilités de Saint-Flour XXXI – 2001. Berlin: Springer (ISBN 3-540-20832-1/pbk). Lect. Notes Math. 1837, 191-312 (2004).

This text is based on a St. Flour lecture course the author delivered in 2001, but contains additional material. Random walk in random environment (RWRE) is a stochastic process with two sources of randomness: first, the environment, which is randomly chosen but kept fixed throughout the time evolution, and, second, the random walk, which is, for fixed environment, a time-homogeneous Markov chain with transition probabilities given by the environment. These processes often differ substantially from simple random walks in an “averaged” environment. Homogenization techniques may fail, due to “unusual pockets” or “traps” in the environment. The first part of the text deals with RWRE on \(Z\). This model is rather well understood even for a non-i.i.d. environment. For one-dimensional nearest-neighbour RWRE, the following topics are considered: transience/recurrence, the law of large numbers (i.e. the speed of the RWRE), central limit theorems, large deviations, subexponential tail asymptotics, non-standard limit laws and aging properties for the recurrent case. Some of the proofs are not limited to the one-dimensional case: for example, the author explains the method of the “environment viewed from the particle” which has also been used in different contexts.
The second part of the text treats RWRE on \(Z^d\) for \( d>1\). Here, much less is known than for the one-dimensional case, but there has been considerable progress in recent years. The author presents results on \(0-1\)-laws and on the law of large numbers. He describes regeneration ideas which have played a major role in the field, and which allow to relax the assumption of an i.i.d. environment. Further topics are the central limit theorem for balanced RWRE, large deviations for nestling walks and Kalikow’s condition which allows to describe a “ballistic” regime. The text contains an extensive list of references. As the author points out in the introduction, multidimensional RWRE is a very active research area, and after the text has appeared, impressing CLT results have been obtained recently [see A. S. Sznitman and O. Zeitouni, “An invariance principle for isotropic diffusions in random environment” (Preprint)].
For the entire collection see [Zbl 1034.60001].

Reviewer: Nina Gantert (Münster)

MSC:

60K37	Processes in random environments
60F10	Large deviations
60G50	Sums of independent random variables; random walks
82B41	Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
82B44	Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics

Keywords:

random walk in random environment; central limit theorem; ergodic theorems; large deviations