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

$S$ \left(M\left(z\right)\right)$, where entries of the linear operator $ $S$ : $C$ ^{N\times N} \rightarrow $C$ ^{N\times N} $d e c a y e x p o n e n t i a l l y . W e s h o w t h a t$ M(z)$ also has exponential off-diagonal decay and can be represented as Laurent series with coefficients determined by entries of $S$ . 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.

We consider the solution of Matrix Dyson Equation $-M\left(z\right)^{-1} = z

$S$ \left(M\left(z\right)\right)$, where entries of the linear operator $ $S$ : $C$ ^{N\times N} \rightarrow $C$ ^{N\times N} $d e c a y e x p o n e n t i a l l y . W e s h o w t h a t$ M(z)$ also has exponential off-diagonal decay and can be represented as Laurent series with coefficients determined by entries of $S$ . 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.

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