  We provide a proof that the stationary macroscopic current of particles in a
random lattice Lorentz gas satisfies Fick's law when connected to particles
reservoirs. We consider a box on a d+1 dimensional lattice and when $d\geq7$,
we show that under a diffusive rescaling of space and time, the probability to
find a current different from its stationary value is exponentially small in
time. Its stationary value is given by the conductivity times the difference of
chemical potentials of the reservoirs. The proof is based on the fact that in
high dimension, random walks have a small probability of making loops or
intersecting each other when starting sufficiently far apart.
