Non-symmetric vector dyson equations

We study the vector Dyson equation $$-\frac{1}{m(z)}=z\mathbf{1}+\mathbf{a}+Sm(z),$$ with parameter $z$ in the complex upper half-plane $\mathbb{C}_+$, where $\mathbf{a}\in\mathbb R^d$ and $S$ is a nonnegative matrix, not necessarily symmetric. This equation has a unique vector solution $m(z)\in\mathbb{C}_+^d$, for which we establish a complete measure decomposition and prove regularity. We then develop a graph-theoretic approach to the singularity and stability problem for non-symmetric matrices $S$. The graph structure of $S$ identifies the possible degeneracies of the stability operator as $z$ approaches the real axis. In particular, for non-backtracking matrices, we prove square-root growth at regular edges, cubic-root growth at regular cusps, and complete stability estimates. We also obtain the corresponding estimates for symmetric matrices in the periodic setting.