Finite-dimensional approximation for the diffusion coefficient in the simple exclusion process

We show that for the mean zero simple exclusion process in ${\mathbb{Z}}^d$ and for the asymmetric simple exclusion process in ${\mathbb{Z}}^d$ for $d\ge 3$, the self-diffusion coefficient of a tagged particle is stable when approximated by simple exclusion processes on large periodic lattices. The proof depends on a similar stability property of the Sobolev inner product associated with the operator.