  In this paper we develop and apply methods for the spectral analysis of
non-self-adjoint tridiagonal infinite and finite random matrices, and for the
spectral analysis of analogous deterministic matrices which are pseudo-ergodic
in the sense of E.B.Davies (Commun. Math. Phys. 216 (2001), 687-704). As a
major application to illustrate our methods we focus on the "hopping sign
model" introduced by J.Feinberg and A.Zee (Phys. Rev. E 59 (1999), 6433-6443),
in which the main objects of study are random tridiagonal matrices which have
zeros on the main diagonal and random $\pm 1$'s as the other entries. We
explore the relationship between spectral sets in the finite and infinite
matrix cases, and between the semi-infinite and bi-infinite matrix cases, for
example showing that the numerical range and $p$-norm $\eps$-pseudospectra
($\eps>0$, $p\in [1,\infty]$) of the random finite matrices converge almost
surely to their infinite matrix counterparts, and that the finite matrix
spectra are contained in the infinite matrix spectrum $\Sigma$. We also propose
a sequence of inclusion sets for $\Sigma$ which we show is convergent to
$\Sigma$, with the $n$th element of the sequence computable by calculating
smallest singular values of (large numbers of) $n\times n$ matrices. We propose
similar convergent approximations for the 2-norm $\eps$-pseudospectra of the
infinite random matrices, these approximations sandwiching the infinite matrix
pseudospectra from above and below.
