Spectral gap for the zero range process with constant rate

We solve an open problem concerning the relaxation time (inverse spectral gap) of the zero range process in ${\mathbf{Z}}^d\mathrm{/}L{\mathbf{Z}}^d$ with constant rate, proving a tight upper bound of $O((\rho +1{)}^2{L}^2)$, where $\rho$ is the density of particles.