Spectral gap for zero-range processes with jump rate g(x)=xγg(x)=xγ

We consider zero-range processes with jump rate g(x)=xγg(x)=xγ for 0<γ≤10<γ≤1. We obtain that for the local process confined to a cube in Zd of width nn, the spectral gap is bounded below by positive multiple of (n2(1+ρ)1−γ)−1(n2(1+ρ)1−γ)−1, where ρ=k/(2n+1)dρ=k/(2n+1)d and kk is the total number of particles in the cube.