A Quantitative Central Limit Theorem for the Effective Conductance on the Discrete Torus

Abstract

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 → ∞. 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 .

The additive structure of elliptic homogenization

“…Versions of (11.3) have been proved in [34,37,8,35,18], with the quantity J replaced by spatial averages of the energy density of the correctors and approximations to the corrector. A version of (11.5) was proved in [31,30], while a version of (11.6) with u r (z) replaced by ∫ Φz,r (u − E[u]) was proved in [24].…”

“…In particular, they were the first to obtain estimates for the correctors at the critical scalings, albeit with suboptimal stochastic integrability (typically finite moment bounds) and with somewhat restrictive ergodic assumptions. Later, central limit theorems for the spatial averages of the gradients and the energy densities of the correctors were obtained using these techniques [34,31,30,18]. These important and influential results were the first to give a complete quantitative picture of the behavior of the first-order correctors on any stochastic model, and have inspired a huge amount of subsequent research.…”

The Choice of Representative Volumes in the Approximation of Effective Properties of Random Materials

“…As a by-product, our work also provides a proof of quantitative normal approximation for a RVE in a different setting than available in the literature so far: To the best of our knowledge, the results on quantitative normal approximation for a RVE in the literature always rely on an assumption that the coefficient field a is obtained as a function of iid random variables [37,50,75] or that the probability distribution of a is subject to a second-order Poincaré inequality like in [36]. In contrast, our result holds under the assumption of finite range of dependence, in which to the best of our knowledge only a qualitative normal approximation result had been known [6].…”

Homogenization of a Random Walk on a Graph in $\mathbb R^d$: An Approach to Predict Macroscale Diffusivity in Media with Finescale Obstructions and Interactions

“…or stationary and ergodic jump rates, our work is restricted to the simpler periodic case. We believe that, by exploiting the ideas that connect the periodic case with the stationary and ergodic case [23,12], the results of this manuscript can be extended to the stationary and ergodic case.…”