Flowing to Normality and the Fate of the Single Ring Theorem
Abstract
Random non-hermitian matrix ensembles with double-sided rotation invariance obey, in the limit of large matrix size, the Single Ring Theorem, which states that the support of the mean eigenvalue distribution in the complex plane is either a disk or an annulus. In contrast, rotational-invariant random normal matrix ensembles can have mean eigenvalue densities supported over any number of concentric annuli in the complex plane. In this paper we introduce and investigate, both analytically and numerically, a non-hermitian matrix model which flows from a generic matrix distribution obeying the Single Ring Theorem to a distribution of normal matrices by tuning a parameter which penalizes non-normality. We observe numerically breakdown of the Single Ring Theorem as the model flows towards normality, and determine the critical value of the parameter at which the transition occurs. We also study in detail the behavior of the singular values of these matrices under the flow. These singular values form a Fermi gas confined to the positive half-line. In particular, we find that at small values of the flow parameter, the interparticle spacings in the gas exhibit Wigner-Dyson repulsion, whereas for asymptotically large values of the flow parameter, at the normal matrix endpoint of the flow, the spacing statistics is Poissonian. The flow interpolates continuously between these two types of statistics. However, this change in statistics is not related directly to breaking of the Single Ring Theorem, which occurs very early-on along the flow, in the regime of Wigner-Dyson statistics. Finally, we introduce a certain ensemble of random permutations associated with the gas, and make a conjecture on how to use it in order to reconstruct approximately the average density of complex eigenvalues from that of the singular values in the large-N limit.
Versions
Related
- A note on the asymmetric discretised ring theorem
- The Hilbert Basis Theorem for Non-Associative Skew Power Series Rings and Non-Associative Skew Laurent Series Rings over a Non-Associative Coefficient Ring
- Invariant Sphere Theorem and Ring-Coupled Systems
- A single generation theorem for simple unital C*-algebras with CPoU and comparison
- A single induction proof of Simons' Riemannian holonomy theorem
BibTeX
@misc{feinberg2026flowing,
title = {{Flowing to Normality and the Fate of the Single Ring Theorem}},
author = {Joshua Feinberg and Roman Riser and Richard Scalettar and Anthony Zee},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.15791},
eprint = {2606.15791},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.15791},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Flowing to Normality and the Fate of the Single Ring Theorem
AU - Joshua Feinberg
AU - Roman Riser
AU - Richard Scalettar
AU - Anthony Zee
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2606.15791
UR - https://arxiv.org/abs/2606.15791
AB - Random non-hermitian matrix ensembles with double-sided rotation invariance obey, in the limit of large matrix size, the Single Ring Theorem, which states that the support of the mean eigenvalue distribution in the complex plane is either a disk or an annulus. In contrast, rotational-invariant random normal matrix ensembles can have mean eigenvalue densities supported over any number of concentric annuli in the complex plane. In this paper we introduce and investigate, both analytically and numerically, a non-hermitian matrix model which flows from a generic matrix distribution obeying the Single Ring Theorem to a distribution of normal matrices by tuning a parameter which penalizes non-normality. We observe numerically breakdown of the Single Ring Theorem as the model flows towards normality, and determine the critical value of the parameter at which the transition occurs. We also study in detail the behavior of the singular values of these matrices under the flow. These singular values form a Fermi gas confined to the positive half-line. In particular, we find that at small values of the flow parameter, the interparticle spacings in the gas exhibit Wigner-Dyson repulsion, whereas for asymptotically large values of the flow parameter, at the normal matrix endpoint of the flow, the spacing statistics is Poissonian. The flow interpolates continuously between these two types of statistics. However, this change in statistics is not related directly to breaking of the Single Ring Theorem, which occurs very early-on along the flow, in the regime of Wigner-Dyson statistics. Finally, we introduce a certain ensemble of random permutations associated with the gas, and make a conjecture on how to use it in order to reconstruct approximately the average density of complex eigenvalues from that of the singular values in the large-N limit.
ER -