From quenched invariance principle to semigroup convergence with applications to exclusion processes

Abstract

Consider a random walk on ${\mathbb{Z}}^d$ in a translation-invariant and ergodic random environment and starting from the origin. In this short note, assuming that a quenched invariance principle for the opportunely-rescaled walks holds, we show how to derive an $L^1$-convergence of the corresponding semigroups. We then apply this result to obtain a quenched pathwise hydrodynamic limit for the simple symmetric exclusion process on ${\mathbb{Z}}^d$, $d\ge 2$, with i.i.d. symmetric nearest-neighbors conductances ${\mathit{\omega }}_{xy}\in [0,\mathrm{\infty })$ only satisfying

$$\mathbb{Q}({\mathit{\omega }}_{xy}>0)>p_c,$$

where $p_c$ is the critical value for bond percolation.