  Following Barany et al., who proved that large random lattice zonotopes
converge to a deterministic shape in any dimension after rescaling, we
establish a central limit theorem for finite-dimensional marginals of the
boundary of the zonotope. In dimension 2, for large random convex lattice
polygons contained in a square, we prove a Donsker-type theorem for the
boundary fluctuations, which involves a two-dimensional Brownian bridge and a
drift term that we identify as a random cubic curve.
