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

TL;DR

This paper proves a quantitative central limit theorem for the effective conductance on a discrete torus, showing it behaves like an average of conductances with precise variance asymptotics.

Contribution

It provides the first precise asymptotic description of the variance of the effective conductance in a random conductance model on a discrete torus.

Findings

Effective conductance follows a CLT as system size grows.

Variance of effective conductance is asymptotically characterized.

Conductance behaves like a spatial average with logarithmic corrections.

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 $A_L$.