  We study the nature and mechanisms of broken ergodicity (BE) in specific
random walk models corresponding to diffusion on random potential surfaces, in
both one and high dimension. Using both rigorous results and nonrigorous
methods, we confirm several aspects of the standard BE picture and show that
others apply in one dimension, but need to be modified in higher dimensions.
These latter aspects include the notions that at fixed temperature confining
barriers increase logarithmically with time, that ``components'' are
necessarily bounded regions of state space which depend on the observational
timescale, and that the system continually revisits previously traversed
regions of state space. We examine our results in the context of several
experiments, and discuss some implications of our results for the dynamics of
disordered and/or complex systems.
