  We study fluctuation properties of a one-dimensional anisotropic Heisenberg
model out of equilibrium, focusing in particular on the gapped regime. Within
the open-system setting we study large-deviation properties of the spin
current. Numerically evaluating the first four current cumulants in a
nonequilibrium stationary state at high energies, we find that the first two
cumulants scale with the system size in a diffusive way, while the 3rd and the
4th cumulants do not. This means that the model is not an ordinary diffusive
spin conductor.
