At criticality for long-range jumps with i.i.d. elliptic conductances, the model homogenizes to the Laplacian at rate 1/sqrt|ln ε|, implying sqrt(t log t) quenched scaling to Brownian motion with diffusivity given solely by the mean conductance.
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Quantitative homogenization for the critical long-range random conductance model
At criticality for long-range jumps with i.i.d. elliptic conductances, the model homogenizes to the Laplacian at rate 1/sqrt|ln ε|, implying sqrt(t log t) quenched scaling to Brownian motion with diffusivity given solely by the mean conductance.