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

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