Localization in One Dimension

Random band matrices are Hermitian matrices with random entries supported in a band of width W around the diagonal. The eigenfunctions of such matrices are expected to decay exponentially at the scale W^2 in dimension one, and exp(CW^2) in dimension two. Remarkably, the same scaling is expected for the cycle lengths in various models of random permutations where points are typically displaced by distances of order W. In this talk I will discuss some recent progress on these problems and some open questions.