Quantitative stochastic homogenization for long-range random walks with critical jump index

In this paper, we study the stochastic homogenization for a class of symmetric random walks in random conductance model, whose one-step transition probability from $x$ to $y$ is proportional to $|x-y|^{-d-2}$. As the associated jumping kernel fails to be $L^2$-integrable yet admits a finite $\alpha$-th moment for all $\alpha\in (0,2)$, we refer to the corresponding process $(X^\w_t)_{t\ge0}$ as a long-range random walk with critical jump index. In this critical regime, the scaled process $\bigl(k^{-1}X_{k^2(\log k)^{-1}t}\bigr)_{t\ge 0}$, whose scaling order is different from the diffusive scaling and the $\alpha$-stable scaling, converges to a Brownian motion. Besides characterizing the limiting Brownian motion, we will give a convergence rate for associated scaled resolvents, which obeys the order $(\log k)^{-\frac{1}{2}+\frac{1}{2(d-2)}+\varepsilon}$ with any $\varepsilon>0$ for all $d>3$.