  We prove the existence of localized states at the edges of the bands for the
two-dimensional Landau Hamiltonian with a random potential, of arbitrary
disorder, provided that the magnetic field is sufficiently large. The
corresponding eigenfunctions decay exponentially with the magnetic field and
distance. We also prove that the integrated density of states is Lipschitz
continuous away from the Landau energies. The proof relies on a Wegner estimate
for the finite-area magnetic Hamiltonians with random potentials and
exponential decay estimates for the finite-area Green's functions. The proof of
the decay estimates for the Green's functions uses fundamental results from
two-dimensional bond percolation theory.
