It is known that many equations of interest in Mathematical Physics display solutions which are only asymptotically invariant under transformations ( e. g. scaling and/or translations) which are not symmetries of the considered equation. In this note we extend the approach to asymptotic symmetries for the analysis of PDEs, to the case of difference equations.
Gaeta, G., Levi, D., Mancinelli, R. (2005). Asymptotic symmetries of difference equations on a lattice. JOURNAL OF NONLINEAR MATHEMATICAL PHYSICS, 12, 137-146 [10.2991/jnmp.2005.12.Supplement 2.9].