This research deals with the long term behavior of RWRE (random walks in random environments). More precisely, one looks at random walks on the d-dimensional lattice, whose transition probabilities form a random field indexed by the vertices of the lattice. Not only is the RWRE providing a serious mathematical challenge, many of the results proved in this field, especially in the one dimensional case, stand in sharp contrast with the behavior of classical random walk. For example, in dimension 1, subdiffusive behavior and aging phenomena are present, whereas in high dimension one may construct RWREs with nonzero (averaged) local drift pointing in one direction while the walk drifts away (with positive speed) in the opposite direction. Some of the mathematical challenge emanates from the fact that in the multidimensional case, conditioned on the transition probabilities random field, the RWRE is not reversible. This precludes the use of much of the available theory of homogenization, and requires the development of new techniques, most notably through the use of regeneration times and of renormalization ideas. In spite of rapid progress that was achieved in the last few years by several researchers including Sznitman, Varadhan, and Zerner, many fundamental questions remain unanswered. Among these one notes the existence of 0-1 laws, laws of large numbers in full generality, transience and recurrence criteria, the relation between quenched and annealed large deviations behavior, the existence (or non-existence) of sub-diffusive behavior for high dimension, and the behavior of RWREs in mixing (as opposed to i.i.d.) environments. Random walks are arguably the stochastic processes most studied by mathematicians, having the widest range of applications in fields as diverse as the physical sciences, engineering, and the social sciences. Though the theory of random walks is by now well developed, this is not at all the case when one changes the medium in which the walk evolves to a random medium, thus obtaining a random walk in random environment (RWRE). Such RWREs can be used to model many problems of motion in random media in the physical and engineering sciences, and are mathematically appealing because on the one hand the model is very simply stated, while on the other hand established tools for studying processes in random media are not applicable in the study of RWRE. The goal of the current proposal is to develop new basic probabilistic techniques that will allow one to make provable predictions concerning the behavior of RWRE. It is expected that such techniques will be useful in the study of other processes, and link naturally to the study of trapping models and reinforced random walks. While not explicitly targeted in this proposal, trapping models have recently been used to study the environmental impact of nuclear waste, and RWREs can naturally be used in this context to model the spread of contamination in a real environment, once the required mathematical background and tools are in place.
