Topological insulators are materials characterized by a non-trivial topological invariant. A striking feature is the presence of edge states at the interface between two topogically distinct materials that propagate in a privileged direction. Since this asymmetric transport is protected topologically, it is immune to random fluctuations that do not break the invariant. The resulting absence of back-scattering makes these materials appealing practically.

We model such two-dimensional materials by systems of Dirac equations. Topological invariants in the form of Fredholm indices can be assigned to these systems to describe the asymmetric edge states. We develop a scattering theory to assess the quantitative influence of random fluctuations on transport along the edge. In the diffusive regime, we obtain a configuration where transport and (Anderson) localization coexist: a number of modes prescribed by topology transport while all other modes localize. The results also generalize to the setting of fermionic time reversal symmetry, where the Fredholm index is replaced by a ${\mathbb Z}_2$ index.