Quenched scaling limit for biased random walks on random, heavy tailed conductances: low dimensions

We consider a random walk amongst positive random conductances on $\mathbb{Z}^d, d \ge 2$, with directional bias. When the conductances have a stable distribution with parameter $\gamma \in (0, 1)$, the walk is sub-ballistic. In this regime Fribergh and Kious (Ann. Prob. 2018) derived an annealed scaling limit for the appropriately rescaled walk towards the Fractional Kinetics process. We prove the quenched version of this result for all $d \ge 2$.

Umberto De Ambroggio, Carlo Scali · 2025