Abstract. We study the behavior at infinity of solutions of discrete Schrödinger equations. First we study pairs of discrete Schrödinger equations whose potential functions differ by a quantity that can be considered small in a suitable sense as the index n → ∞ . With simple assumptions on the growth rate of the solutions of the original system, we show that the perturbed system has a fundamental set of solutions with the same behavior at infinity, employing a variation-ofconstants scheme to produce a convergent iteration for the solutions of the second equation in terms of those of the original one.Next, we present a sharp discrete analogue of the Liouville-Green (JWKB) transformation, making it possible to derive exponential behavior at infinity of a single difference equation, by explicitly constructing a comparison equation to which the perturbation results apply. After that we use the relations between the solution sets of two discrete Schrödinger equations differing by a perturbation to derive exponential dichotomy of solutions and to elucidate the structure of transfer matrices.A final section contains illustrative examples, including some with large, oscillatory potentials, and an appendix discusses the connection between the discrete Schrördinger problem and orthogonal polynomials on the real line.Mathematics subject classification (2010): Primary 34E10, 34L40, 39A06, 39A22; Secondary 39C41.