A wanderer strolling along an unfamiliar scenery faces a dilemma each time he reaches an intersection. Which way should she turn? If the wanderer uses a random die in order to decide on its next move, is she assured to return to its starting point? Surprisingly, this question relates to transport properties in (homogeneous) material, to heat conductance of material, and to the analysis of multi-agent networks.
If the die used is biased in different ways in different location, which we refer to as "disorder", one expects that as long as the bias is small and evenly distributed across location, the path of the wandered will be similar to that in the unbiased case. In a series of work, WIS researchers, in collaboration with others, showed that this is not the case in general (there are ways to maliciously modify the distribution of the bias and create very different transport properties). On the other hand, under appropriate isotropy assumptions, namely, that the configuration of biases looks similar in any direction that the wanderer chooses, they were able to demonstrate (in dimension three and larger) that for small disorder, the behavior of the walk is the same as in the case were there is no disorder.