Spectral gap for zero-range processes with jump rate

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