Anomalous singularity of the solution of the vector Dyson equation in the critical case

Abstract

The vector Dyson equation (VDE) is an equation of the form −1/m = z + Sm, where S is a fixed square matrix, z lies in the complex upper half-plane, and m is a vector-function of z. We consider the solution of the VDE in the case when S has a block staircase structure with (n − 1) different critical zero blocks below the strictly positive anti-diagonal and all elements right above the anti-diagonal are strictly positive. We prove that the components of m behave as fractional powers of z in the neighborhood of zero and show that the self-consistent density of states ρ(E) behaves as |E|−n−1n+1 as E tends to zero, where n2 is the number of blocks. Both constant block and non-constant block cases are considered. In the non-constant case, uniform estimates for the components of m are obtained.