Flat band states: Disorder and nonlinearity

We study the critical behavior of Anderson localized modes near intersecting flat and dispersive bands in the quasi-one-dimensional diamond ladder with weak diagonal disorder W. The localization length xi of the flat band states scales with disorder as xi similar to W-gamma, with gamma approximate to 1.3, in contrast to the dispersive bands with gamma = 2. A small fraction of dispersive modes mixed with the flat band states is responsible for the unusual scaling. Anderson localization is therefore controlled by two different length scales. Nonlinearity can produce qualitatively different wave spreading regimes, from enhanced expansion to resonant tunneling and self-trapping.