  We study various qualitative and quantitative (global) unique continuation
properties for the fractional discrete Laplacian. We show that while the
fractional Laplacian enjoys striking rigidity properties in the form of
(global) unique continuation properties, the fractional discrete Laplacian does
not enjoy these in general. While discretization thus counteracts the strong
rigidity properties of the continuum fractional Laplacian, by discussing
quantitative forms of unique continuation, we illustrate that these properties
can be recovered if exponentially small (in the lattice size) correction terms
are added. This in particular allows us to deduce uniform stability properties
for a discrete, linear inverse problem for the fractional Laplacian. We
complement these observations with a transference principle and the discussion
of these properties on the discrete torus.
