Abstract
We 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. We 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 ad-dimensional walker moving in a symmetric random environment under very mild assumptions on the distribution of the environment, (iii) for a tagged particle in ad-dimensional symmetric lattice gas which allows interchanges, (iv) for a tagged particle in ad-dimensional system of interacting Brownian particles. Our formulation also leads naturally to bounds on the diffusion constant.
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De Masi, A., Ferrari, P.A., Goldstein, S. et al. An invariance principle for reversible Markov processes. Applications to random motions in random environments. J Stat Phys 55, 787–855 (1989). https://doi.org/10.1007/BF01041608
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DOI: https://doi.org/10.1007/BF01041608