Consider a nearest neighbor random walk on with discrete time. Imagine that there is a label at each vertex of that gives the probability that the random walk will move to the right. Such a sequence of labels is an environment for a random walk on . If the labels are chosen at random, we have a random walk on in a random environment.
We restrict ourselves to environments in which the random walk escapes to the right with positive speed almost surely, and we study how the speed depends on the environment. Provided that the environment is an ergodic measure-preserving system, there is an explicit expression for the speed. The result has a very appealing interpretation in terms of electric networks.
We introduce the branching number of an environment, which, roughly speaking, measures the extent to which the random walk prefers the positive direction to the negative. Then we compare a random walk on in a random, i.e. disordered, environment to a random walk on in a constant environment with the same branching number and ask: Is the random walk slowed down by the disorder in the environment? The next question is: Is the random walk slowed down more by more disorder in the environment? The answer will be given via Markov chain environments with the same branching number but varying entropy.
The answers can also be found in a paper to appear in Journal of Theoretical Probability.