Determining the spectral type of discrete Schroedinger operators related to
lattice gauge fields is open in many examples.
Such operators exist because some
ergodic cohomology groups are trivial.
An example of this class of operators is obtained, when the magnetic flux on
the plaquette are independent, identically distributed random
variables and where the lattice gauge fields are in general not
independent random variables.
An example, where the spectral properties are already not known
is a two dimensional lattice with a IID magnetic field with uniform
distribution on U(1). Experiments indicate that there are
in this case.