Solutions of the vector difference equationy(x+e)−y(x−e)=2ef(x,y(x)),x being a complex variable and e>0 a small parameter, are constructed that are analytic onx-domains Ω which are independent of e. As the first case, horizontally convex bounded domains are considered, i.e., domains having the property that for eachx, x′∈Ω with the same imaginary part, the interval [x, x′] is contained in Ω; also considered are unbounded domains such as sectors open on the left or on the right. Using these results, it is shown that the Hausdorff distance between separatrices of certain systems of difference equations is exponentially small with respect to e. As an application, the so-called ghost solutions of the discretized logistic equation are considered in detail and, in particular, the lengths of the levels are estimated. Other applications, e.g., to the standard mapping, are presented.