  This article is devoted to the analysis of a Monte Carlo method to
approximate effective coefficients in stochastic homogenization of discrete
elliptic equations. We consider the case of independent and identically
distributed coefficients, and adopt the point of view of the random walk in a
random environment. Given some final time t>0, a natural approximation of the
homogenized coefficients is given by the empirical average of the final squared
positions re-scaled by t of n independent random walks in n independent
environments. Relying on a quantitative version of the Kipnis-Varadhan theorem
combined with estimates of spectral exponents obtained by an original
combination of PDE arguments and spectral theory, we first give a sharp
estimate of the error between the homogenized coefficients and the expectation
of the re-scaled final position of the random walk in terms of t. We then
complete the error analysis by quantifying the fluctuations of the empirical
average in terms of n and t, and prove a large-deviation estimate, as well as a
central limit theorem. Our estimates are optimal, up to a logarithmic
correction in dimension 2.
