Quantitative stochastic homogenization for random conductance models with stable-like jumps

We consider random conductance models with long range jumps on ${\mathbb{Z}}^d$, where the one-step transition probability from $x$ to $y$ is proportional to $w_{x,y}\mathrm{\mid }x-y{\mathrm{\mid }}^{-d-\alpha }$ with $\alpha \in (0,2)$. Assume that $\{w_{x,y}{\}}_{(x,y)\in E}$ are independent, identically distributed and uniformly bounded non-negative random variables with $\text{Ee}{w}_{x,y}=1$, where $E$ is the set of all unordered pairs on ${\mathbb{Z}}^d$. We obtain a quantitative version of stochastic homogenization for these random walks, with explicit polynomial rates up to logarithmic corrections.