From quenched invariance principle to semigroup convergence with applications to exclusion processes
Alberto Chiarini, Simone Floreani, Federico Sau
TL;DR
The paper shows that a quenched invariance principle (QIP) from the origin, together with translation-invariance and ergodicity, yields an $L^1$ convergence of semigroups across starting points. This semigroup convergence then enables a quenched pathwise hydrodynamic limit for the symmetric exclusion process on the infinite percolation cluster with i.i.d. conductances, leading to a heat equation with a diffusion coefficient $\sigma^2$ (scaled by the cluster density $q$). The approach unifies random-walk scaling limits with hydrodynamic limits in random environments, and applies to both diffusive and certain sub-diffusive regimes (including Fractional Kinetics) depending on the conductance tail. The results provide robust, environment-averaged hydrodynamic behavior for IPS in complex media and offer a blueprint for analyzing other exclusion-type systems under randomness. Practically, this advances understanding of diffusion in disordered media and informs models where transport occurs on random substructures like percolation clusters.
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 $ω_{xy}\in [0,\infty)$ only satisfying $$\mathbb{Q}(ω_{xy}>0)>p_c\ ,$$ where $p_c$ is the critical value for bond percolation.