  We consider the asymmetric zero range process in dimensions $d \geq 2$.
Assume the initial density profile is a perturbation of the constant density,
which has order $N^{-\alpha}$, $\alpha \in (0,1)$, and is constant along the
drift direction. Here, $N$ is the scaling parameter. We show that under some
constraints on the jump rate of the zero range process, the perturbed quantity
macroscopically obeys the heat equation under diffusive scaling.
