Hydrodynamic behavior of 1D subdiffusive exclusion processes with random conductances

Consider a system of particles performing nearest neighbor random walks on the lattice ${\mathbb{Z}}$ under hard-core interaction. The rate for a jump over a given bond is direction-independent and the inverse of the jump rates are i.i.d. random variables belonging to the domain of attraction of an α-stable law, 0 < α < 1. This exclusion process models conduction in strongly disordered 1D media. We prove that, when varying over the disorder and for a suitable slowly varying function L, under the super-diffusive time scaling N ^{1 +1/α} L(N), the density profile evolves as the solution of the random equation ${\partial_t \rho = \mathfrak{L}_W \rho}$ , where ${\mathfrak{L}_W}$ is the generalized second-order differential operator ${\frac d{du} \frac d{dW}}$ in which W is a double-sided α-stable subordinator. This result follows from a quenched hydrodynamic limit in the case that the i.i.d. jump rates are replaced by a suitable array ${\{\xi_{N,x} : x\in\mathbb{Z}\}}$ having same distribution and fulfilling an a.s. invariance principle. We also prove a law of large numbers for a tagged particle.