Disk-annulus transition and localization in random non-Hermitian tridiagonal matrices

Eigenvalues and localization of eigenvectors of non-Hermitian tridiagonal periodic random matrices are studied by means of the Hatano-Nelson deformation. The support of the spectrum undergoes a disk to annulus transition, with inner radius measured by the complex Thouless formula. The inner bounding circle and the annular halo are stuctures that correspond to the two-arcs 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. This disk-annulus transition is reminiscent of Feinberg and Zee's transition observed in full complex random matrices.