Scaling limit of the complex mobility matrix for the random conductance model on ${\mathbb{T}}_N^d$

We consider a continuous-time random walk on the $d$-dimensional torus ${\mathbb{T}}_N^d={\mathbb{Z}}^d\mathrm{/}N{\mathbb{Z}}^d$, possibly with long-range, but finite, jumps. The law of the jumps is regulated by a random environment $\xi$ yielding a stationary and ergodic field of random conductances. The complex mobility matrix ${\sigma }_N^{\xi }(\omega )$ measures the linear response of the random walk to a $\cos (\omega t)$-type oscillating external field. By investigating the homogenization properties of the medium, and assuming in addition that the conductances have finite second moment, we show that, for almost every realization of the environment $\xi$, the complex mobility matrix ${\sigma }_N^{\xi }(\omega )$ converges as $N\to +\mathrm{\infty }$ to a deterministic limiting matrix $\sigma (\omega )$ and provide different characterizations of $\sigma (\omega )$.