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

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