Quantitative homogenization for the critical long-range random conductance model

We consider the long-range random conductance model on $\mathbb{Z}^d$ at the critical exponent: the jump rate between sites $x$ and $y$ decays as $\mathbf{a}(x,y) |x-y|^{-(d+2)}$, where $\mathbf{a}(x,y)$ are i.i.d. uniformly elliptic conductances. Below the critical exponent $(d+2)$ the walk converges to a stable process; above it, to Brownian motion with diffusive $\sqrt{t}$ scaling. At criticality the second moment of the jump kernel diverges logarithmically. We establish quantitative homogenization of the associated elliptic equation to the Laplacian at the rate $1/\sqrt{|\ln\varepsilon|}$. As a consequence, we deduce quenched convergence of the random walk to Brownian motion under the anomalous $\sqrt{t \log t}$ scaling. Unlike in standard homogenization, the effective diffusivity is determined by the mean conductance alone, with no corrector contribution at leading order.