Large deviations for the symmetric simple exclusion process in dimensions d >= 3

We consider symmetric simple exclusion processes with L = <(rho)over bar>N- d particles in a periodic d-dimensional lattice of width N. We perform the diffusive hydrodynamic scaling of space and time. The initial condition is arbitrary and is typically far away form equilibrium. It specifies in the s caling limit a density profile on the d-dimensional torus. We are intereste d in the large deviations of the empirical process, N-d[Sigma(I)(L) delta(x i(.))] as random variables taking values in the space of measures on D[0.1] . We prove a large deviation principle, with a rate function that is more o r less universal, involving explicity besides the initial profile, only suc h canonical objects as bulk and self diffusion coefficients.