  We study the current of particles that move independently in a common static
random environment on the one-dimensional integer lattice. A two-level
fluctuation picture appears. On the central limit scale the quenched mean of
the current process converges to a Brownian motion. On a smaller scale the
current process centered at its quenched mean converges to a mixture of
Gaussian processes. These Gaussian processes are similar to those arising from
classical random walks, but the environment makes itself felt through an
additional Brownian random shift in the spatial argument of the limiting
current process.
