A quantitative central limit theorem for the effective conductance on the discrete torus

Abstract

We study a random conductance problem on a $d$-dimensional discrete torus of size $L>0$. The conductances are independent, identically distributed random variables uniformly bounded from above and below by positive constants. The effective conductance $A_L$ of the network is a random variable, depending on $L$, and the main result is a quantitative central limit theorem for this quantity as $L\to \infty$. In terms of scalings we prove that this nonlinear nonlocal function $A_L$ essentially behaves as if it were a simple spatial average of the conductances (up to logarithmic corrections). The main achievement of this contribution is the precise asymptotic description of the variance of $A_L$.

37 pages

References in corpus (3)

• An optimal error estimate in stochastic homogenization of discrete elliptic equations
• A new method of normal approximation
• A short survey of Stein's method