  In this paper we give log-convexity properties for solutions to discrete
Schr\"odinger equations with different discrete versions of Gaussian decay at
two different times. For free evolutions, we use complex analysis arguments to
derive these properties, while in a perturbative setting we use a preliminar
log-convexity result in order to get these properties. Then, by proving a
Carleman inequality we conclude, in one of the cases under study, a discrete
version of Hardy's Uncertainty Principle.
