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.
"Current fluctuations of a system of one-dimensional random walks in random environment." Ann. Probab. 38 (6) 2258 - 2294, November 2010. https://doi.org/10.1214/10-AOP537