Non-symmetric vector dyson equations

2026/07/16 by Jiaoyang Huang, Zhonggen Su, Ruizhe Xu
#math.CV #math-ph #math.FA #math.MP #math.PR #math.SP

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Abstract

We study the vector Dyson equation -(1)/(m(z))=z1+a+Sm(z), with parameter z in the complex upper half-plane ℂ₊, where a∈\mathbb R^d and S is a nonnegative matrix, not necessarily symmetric. This equation has a unique vector solution m(z)∈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.

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