Efficient perturbation expansion for disordered systems

It is an old idea [E. Feenberg, Phys. Rev. 74, 206 (1948)] that the perturbation series for the resolvent of a matrix can be rearranged so as to exclude the multiple scattering.


In our work reported at Conference on Applied Physics (Kiev, Ukraine, 2007) we study this technique in detail. We observe that the believed improvement in convergence of the rearranged series is generally not valid, but is valid for sparse matrices, though the complications introduced by the rearrangement make its brute-force use inefficient. Nevertheless the constructive use of the method is possible: for matrices with i.i.d. random variables on their diagonal the rearrangement enables to take the average explicitly (note that some off-diagonal disorder, e.g. random bond model, can be reformulated as diagonal one).