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Non-symmetric vector dyson equations

  • Jiaoyang Huang
  • Zhonggen Su
  • Ruizhe Xu
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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∈ℝd 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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Cite

BibTeXRIS
BibTeX
@misc{huang2026nonsymmetric,
  title = {{Non-symmetric vector dyson equations}},
  author = {Jiaoyang Huang and Zhonggen Su and Ruizhe Xu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2607.16333},
  eprint = {2607.16333},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.16333},
}
RIS
TY  - UNPB
TI  - Non-symmetric vector dyson equations
AU  - Jiaoyang Huang
AU  - Zhonggen Su
AU  - Ruizhe Xu
PY  - 2026
DA  - 2026/10/21
PB  - arXiv
DO  - 10.48550/arxiv.2607.16333
UR  - https://arxiv.org/abs/2607.16333
AB  - 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∈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.
ER  -