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

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