Abstract: We consider an elliptic operator on the discrete d-dimensional lattice whose coefficient matrix is an i.i.d. perturbation of the identity.
Recently, Jean Bourgain introduced novel techniques from harmonic analysis to prove the convergence of the Feshbach-Schur perturbation series for the averaged Green's function of this model. Our main contribution is a refinement of Bourgain's approach which yields a conjecturally nearly optimal decay estimate. As an application, we derive estimates on higher derivatives of the averaged Green's function which go beyond the second derivatives considered by Delmotte-Deuschel and related works.
This is joint work with Jongchon Kim (IAS).