  Eigenvalues and eigenvectors of non-Hermitian tridiagonal periodic random
matrices are studied by means of the Hatano-Nelson deformation. The deformed
spectrum is annular-shaped, with inner radius measured by the complex Thouless
formula. The inner bounding circle and the annular halo are stuctures that
correspond to the two-arc and wings observed by Hatano and Nelson in deformed
Hermitian models, and are explained in terms of localization of eigenstates via
a spectral duality and the Argument principle.
