Solution of Matrix Dyson Equation for Random Matrices with Fast Correlation Decay

We consider the solution of Matrix Dyson Equation $-M\left(z\right)^{-1} = z + \mathcal{S}\left(M\left(z\right)\right)$, where entries of the linear operator $\mathcal{S}: \mathbb{C}^{N\times N} \rightarrow \mathbb{C}^{N\times N}$ decay exponentially. We show that $M(z)$ also has exponential off-diagonal decay and can be represented as Laurent series with coefficients determined by entries of $\mathcal{S}$. We also prove that for Hermitian random matrices with exponential correlation decay empirical density converges to the deterministic density obtained from $M(z)$. These results have already been proved in [arXiv:1604.08188] with the resolvent method, here we give an alternate proof via the conceptually much simpler moment method.