Lifshitz tails, instantons and long time decay for diffusion in media with random traps

In media with random traps the long time survival has a stretched exponential decay, related to a Lifshitz tail in the density of states. We first discuss exact results on this behavior in one dimension, where exact integral equations can be analyzed. Then we show that the stretched exponential behavior is modified when each site contains a random trap. Finally we consider large but finite, three dimensional systems with a small concentration of traps. Here the long time decay is exponential, with a rate that varies from sample to sample. The tail of the distribution function of these rates is discussed.