Superdiffusive scaling limits for the symmetric exclusion process with slow bonds

In [1], the hydrodynamic limit in the diffusive scaling of the symmetric simple exclusion process with a finite number of slow bonds of strength n−β has been studied. Here n is the scaling parameter and β > 0 is fixed. As shown in [1], when β > 1, such a limit is given by the heat equation with Neumann boundary conditions. In this work, we find more non-trivial super-diffusive scaling limits for this dynamics. Assume that there are k equally spaced slow bonds in the system. If k is fixed and the time scale is k2nθ, with θ∈(2,1+β), the density is asymptotically constant in each of the k boxes, and equal to the initial expected mass in that box, i.e., there is no time evolution. If k is fixed and the time scale is k2n1+β, then the density is also spatially constant in each box, but evolves in time according to the discrete heat equation. Finally, if the time scale is k2n1+β and, additionally, the number of boxes k increases to infinity, then the system converges to the continuous heat equation on the torus, with no boundary conditions.