Transport and localization in random media: theory and applications

A variance lower bound for the effective conductivity in random homogenization

James Nolen

Duke University

Consider the effective conductivity matrix for a sample of random material having size $L > 0$, where the random conductivity is stationary and has short-range dependence. As $L$ increases to infinity, the finite-sample effective conductivity $A_L$ converges to a deterministic matrix $A$, the homogenized conductivity coefficient. Gloria and Otto proved an upper bound on the variance of $A_L$ of the order $L^d$ (i.e. the volume), as would be the case for a sum of $L^d$ independent random variables. In this talk, I'll describe a sufficient condition on the coefficient field which implies that an $O(L^d)$ lower bound also holds. Examples will be given, illustrating the use of the sufficient condition for different random conductivity laws. In deriving this result, key ideas are (1) the coupling of laws of the coefficient matrix and (2) understanding the effect of a large inclusion in an otherwise homogeneous background field. This variance lower bound plays a role in normal approximation (via Stein's method) for the random variable $A_L$. This is joint work with Felix Otto.