Spectral gap for zero-range processes with jump rate g(x)=x[gamma]

We consider zero-range processes with jump rate g(x)=x[gamma] for 0<[gamma]<=1. We obtain that for the local process confined to a cube in of width n, the spectral gap is bounded below by positive multiple of (n2(1+[rho])1-[gamma])-1, where [rho]=k/(2n+1)d and k is the total number of particles in the cube.