  There has been much recent interest, initiated by work of the physicists
Hatano and Nelson, in the eigenvalues of certain random non-Hermitian periodic
tridiagonal matrices and their bidiagonal limits. These eigenvalues cluster
along a "bubble with wings" in the complex plane, and the corresponding
eigenvectors are localized in the wings, delocalized in the bubble. Here, in
addition to eigenvalues, pseudospectra are analyzed, making it possible to
treat the non-periodic analogues of these random matrix problems. Inside the
bubble, the resolvent norm grows exponentially with the dimension. Outside, it
grows subexponentially in a bounded region that is the spectrum of the
infinite-dimensional operator. Localization and delocalization correspond to
resolvent matrices whose entries exponentially decrease or increase,
respectively, with distance from the diagonal. This article presents theorems
that characterize the spectra, pseudospectra, and numerical range for the four
cases of finite bidiagonal matrices, infinite bidiagonal matrices ("stochastic
Toeplitz operators"), finite periodic matrices, and doubly infinite bidiagonal
matrices ("stochastic Laurent operators").
