Finite volume approximation of the effective diffusion matrix: The case of independent bond disorder

Abstract

Abstract. Consider uniformly elliptic random walk on Z d with independent jump rates across nearest neighbour bonds of the lattice. We show that the infinite volume effective diffusion matrix can be almost surely recovered as the limit of finite volume periodized effective diffusion matrices.

Approximation of the effective conductivity of ergodic media by periodization

“…We will consider a symmetric random walk on Z d as in [CI01] but with ergodic jump rates instead of i.i.d. The random ergodic environment will be represented by the random d-dimensional vector ξ i (η) (i ∈ {1, .…”

“…It is well known ([KV86], [MFGW89], [CI01]) that in the annealed regime (under the law µ ⊗ P ξ(η) 0 ) as ǫ ↓ 0, ǫX t/ǫ 2 , ξ(η) converges in law towards a Brownian Motion with covariance matrix (effective diffusivity) D(ξ, µ).…”

“…It is well known ( [CI01]) that in the quenched regime (under the law P ξ N (η) 0 ) as ǫ ↓ 0, ǫX t/ǫ 2 , ξ N (η) converges in law towards a Brownian Motion on Z d with covariance matrix (effective diffusivity) σ(ξ N , η) (which is a random matrix on X, depending on the particular realization ξ N (η)).…”

“…The case of the effective diffusivity of a symmetric random walk on Z d , under the condition that its jump rates are i.i.d. has been addressed in [CI01], which also put into evidence an exponential rate of convergence of effective diffusivities of the finite volume approximations of the ergodic medium. It is important to note that the periodic approximation of the effective conductivity of a continuous ergodic operator can be obtained from almost sure G-convergence properties associated to the homogenization of that operator; this is the so called "principle of periodic localization" (equation 5.15, page 155 of [JKO91], [Pia02]).…”

Tail estimates for homogenization theorems in random media

“…The approximation of D 0 by periodic environments is considered in [5,6,22]. Given an environment ω ∈ Ω and an integer N > 1, construct an environment N -periodic on In view of applications to the theory of massless gradient fields on Z d , where periodized states can be used to define the slope-independent surface tension [12], Caputo and Ioffe [6] considered periodic approximations of the homogenized diffusion matrix of a symmetric random walk on Z d with i.i.d.…”