Superdiffusive Scaling Limits for the Symmetric Exclusion Process with Slow Bonds

Abstract

In \cite{fgn1}, the hydrodynamic limit in the diffusive scaling of the symmetric simple exclusion process with a finite number of slow bonds of strength $n^{-\beta}$ has been studied. Here $n$ is the scaling parameter and $\beta>0$ is fixed. As shown in \cite{fgn1}, when $\beta>1$, 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 $k^2n^\theta$, with $\theta\in (2,1+\beta)$, 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 $k^2n^{1+\beta}$, 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 $k^2n^{1+\beta}$ 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.