Equilibrium Perturbations for Asymmetric Zero Range Process under Diffusive Scaling in Dimensions $d\ge 2$

We consider the asymmetric zero range process in dimensions $d\ge 2$. 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.