# Central limit theorem for zero-range processes

J. Quastel, H. Jankowski, J. Sheriff

We consider additive functionals $\int_0^t V(\eta_s) ds$ of symmetric zero-range processes, where $V$ is a mean zero local function. In dimensions $1$ and $2$ we obtain a central limit theorem for $a^{-1}(t) \int_0^t V(\eta_s) ds$ with $a(t) = \sqrt{t\log t}$ in $d=2$ and $a(t) = t^{3/4}$ in $d=1$ and an explicit form for the asymptotic variance $\sigma^2$. The transient case $d\ge 3$ can be handled by standard arguments [KV, SX,S]. We also obtain corresponding invariance principles. This generalizes results obtained by Port (see [CG]) for noninteracting random walks and Kipnis [K] for the symmetric simple exclusion process. Our main tools are the martingale method together with $L^2$ decay estimates [JLQY] for the process semigroup.