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Hard-Edge Determinant Fields and Second-Moment Universality for Random Matrix Ratios

  • Guilherme Vianna
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Abstract

We study the microscopic spectrum at the origin of ratios of independent complex Girko matrices. Under bounded-density and finite-moment assumptions, together with circular second-moment matching, we prove compact-uniform convergence of the normalized determinants to a random entire function constructed from the complex hard edge and an independent Ginibre array. The zero divisor of this limiting determinant field has the law of the infinite complex Ginibre process. Consequently, the first finitely many smallest and largest eigenvalue moduli, the inner and outer spectral radii, and eigenvalue counts in fixed bounded sets have universal limits depending only on the second moments. This answers the second-moment question posed by Chafaï, García-Zelada, and Xu for the spectral radii. We also obtain a multivariate determinant field for finitely many perturbation directions and prove the uniform hard-edge comparison needed to extend the conclusions to triangular arrays of atom laws.

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Cite

BibTeXRIS
BibTeX
@misc{vianna2026hardedge,
  title = {{Hard-Edge Determinant Fields and Second-Moment Universality for Random Matrix Ratios}},
  author = {Guilherme Vianna},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.27641},
  eprint = {2608.27641},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.27641},
}
RIS
TY  - UNPB
TI  - Hard-Edge Determinant Fields and Second-Moment Universality for Random Matrix Ratios
AU  - Guilherme Vianna
PY  - 2026
DA  - 2026/08/27
PB  - arXiv
DO  - 10.48550/arxiv.2608.27641
UR  - https://arxiv.org/abs/2608.27641
AB  - We study the microscopic spectrum at the origin of ratios of independent complex Girko matrices. Under bounded-density and finite-moment assumptions, together with circular second-moment matching, we prove compact-uniform convergence of the normalized determinants to a random entire function constructed from the complex hard edge and an independent Ginibre array. The zero divisor of this limiting determinant field has the law of the infinite complex Ginibre process. Consequently, the first finitely many smallest and largest eigenvalue moduli, the inner and outer spectral radii, and eigenvalue counts in fixed bounded sets have universal limits depending only on the second moments. This answers the second-moment question posed by Chafaï, García-Zelada, and Xu for the spectral radii. We also obtain a multivariate determinant field for finitely many perturbation directions and prove the uniform hard-edge comparison needed to extend the conclusions to triangular arrays of atom laws.
ER  -