  We use the method of bulk-boundary correspondence of topological invariants
to show that disordered topological insulators have at least one delocalized
state at their boundary at zero energy. Those insulators which do not have
chiral (sublattice) symmetry have in addition the whole band of delocalized
states at their boundary, with the zero energy state lying in the middle of the
band. This result was previously conjectured based on the anticipated
properties of the supersymmetric (or replicated) sigma models with WZW-type
terms, as well as verified in some cases using numerical simulations and a
variety of other arguments. Here we derive this result generally, in arbitrary
number of dimensions, and without relying on the description in the language of
sigma models.
