  An analytic quasi-periodic cocycle is a linear cocycle over a fixed ergodic
torus translation of one or several variables, where the fiber action depends
analytically on the base point. Consider the space of all such cocycles of any
given dimension and endow it with the uniform norm. Assume that the translation
vector satisfies a generic Diophantine condition. We prove large deviation type
estimates for the iterates of such cocycles, which, moreover, are stable under
small perturbations of the cocycle. As a consequence of these uniform
estimates, we establish continuity properties of the Lyapunov exponents
regarded as functions on this space of cocycles. This result builds upon our
previous work on this topic and its proof uses an abstract continuity theorem
of the Lyapunov exponents which we derived in a recent monograph. The new
feature of this paper is extending the availability of such results to cocycles
that are identically singular (i.e. non-invertible anywhere), in the several
variables torus translation setting. This feature is exactly what allows us,
through a simple limiting argument, to obtain criteria for the positivity and
simplicity of the Lyapunov exponents of such cocycles. Specializing to the
family of cocycles corresponding to a block Jacobi operator, we derive
consequences on the continuity, positivity and simplicity of its Lyapunov
exponents, and on the continuity of its integrated density of states.
