An invariance principle for reversible Markov processes: applications to random motions in random environments

The authors present an invariance principle for antisymmetric functions of a reversible Markov process which immediately implies convergence to Brownian motion for a wide class of random motions in random environments. They apply it to establish convergence to Brownian motion (i) for a walker moving in the infinite cluster of the two-dimensional bond percolation model, (ii) for a d-dimensional walker moving in a symmetric random environment under very mild assumptions on the distribution of the environment, (iii) for a tagged particle in a d-dimensional symmetric lattice gas which allows interchanges, (iv) for a tagged particle in a d-dimensional system of interacting Brownian particles. Their formulation also leads naturally to bounds on the diffusion constant.