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PreprintCC BY 4.0arXiv

Exact joint eigenvalue densities of non-Hermitian random matrices are Calogero scattering states

  • Zhenyu Xiao
  • Ze Chen
  • Shinsei Ryu
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Abstract

Determining exact joint eigenvalue densities is central to random matrix theory. We solve this long-standing problem for non-Hermitian matrices with transposition symmetry (complex symmetric and complex self-dual) at arbitrary matrix size. Up to a Vandermonde factor, they are scattering-state wave functions of the Calogero model, a line of particles interacting through an inverse-square potential, with the coupling strength set by the symmetry. In contrast to many previously known joint densities, the densities cannot be written as a gas of eigenvalues with pairwise interactions. We further compute the complex level spacing distributions and two-point spectral correlation functions, which carry power-law tails, absent in a Coulomb gas. Our results shed light on the interplay among random matrices, integrability, and symmetry.

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Cite

BibTeXRIS
BibTeX
@misc{xiao2026exact,
  title = {{Exact joint eigenvalue densities of non-Hermitian random matrices are Calogero scattering states}},
  author = {Zhenyu Xiao and Ze Chen and Shinsei Ryu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.00164},
  eprint = {2609.00164},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.00164},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Exact joint eigenvalue densities of non-Hermitian random matrices are Calogero scattering states
AU  - Zhenyu Xiao
AU  - Ze Chen
AU  - Shinsei Ryu
PY  - 2026
DA  - 2026/09/25
PB  - arXiv
DO  - 10.48550/arxiv.2609.00164
UR  - https://arxiv.org/abs/2609.00164
AB  - Determining exact joint eigenvalue densities is central to random matrix theory. We solve this long-standing problem for non-Hermitian matrices with transposition symmetry (complex symmetric and complex self-dual) at arbitrary matrix size. Up to a Vandermonde factor, they are scattering-state wave functions of the Calogero model, a line of particles interacting through an inverse-square potential, with the coupling strength set by the symmetry. In contrast to many previously known joint densities, the densities cannot be written as a gas of eigenvalues with pairwise interactions. We further compute the complex level spacing distributions and two-point spectral correlation functions, which carry power-law tails, absent in a Coulomb gas. Our results shed light on the interplay among random matrices, integrability, and symmetry.
ER  -