A finite dimensional approximation of the effective diffusivity for a symmetric random walk in a random environment

摘要

We consider a nearest neighbor, symmetric random walk on a homogeneous, ergodic random lattice Zd. The jump rates of the walk are independent, identically Bernoulli distributed random variables indexed by the bonds of the lattice. A standard result from the homogenization theory, see [A. De Masi, P.A. Ferrari, S. Goldstein, W.D. Wick, An invariance principle for reversible Markov processes, Applications to random walks in random environments, J. Statist. Phys. 55(3/4) (1989) 787–855], asserts that the scaled trajectory of the particle satisfies the functional central limit theorem. The covariance matrix of the limiting normal distribution is called the effective diffusivity of the walk. We use the duality structure corresponding to the product Bernoulli measure to construct a numerical scheme that approximates this parameter when d⩾3. The estimates of the convergence rates are also provided.