Spectra of Random Polynomial Matrices: the Petaloid Law
Abstract
We study the distribution of the zeros of det PN(z) where PN(z) is a random monic polynomial matrix, i.e., PN(z)=zdI−∑j=0d−1Aj,Nzj for possibly coupled random matrices Aj,N, scaled to have entrywise variance O(1/N). We provide general conditions under which this distribution almost-surely weakly converges as N→∞ to a deterministic measure, depending on just the variance and covariance of the entries of the Aj. This generalizes the circular, elliptic, and semicircle laws, which concern the special case of this question where d=1. Unlike those classical laws, these measures can combine nonuniform two-dimensional densities with singular components supported on curves, producing a variety of petal-shaped regions, inspiring our name ``the petaloid law''. We give explicit formulas for the limiting densities and supports. Under a Gaussianity assumption, we also show that there are almost surely no eigenvalues outside small neighborhoods of the limiting support.
Related
- Universality for the global spectrum of random inner-product kernel matrices in the polynomial regime
- Berry-Esseen bounds for products of random matrices under a 2+δ polynomial moment
- Characteristic polynomial of self-normalized random matrices
- Large deviations for the largest eigenvalue of a polynomial in Gaussian matrices
- Counting Integral Matrices with Given Cubic Characteristic Polynomial
BibTeX
@misc{shah2026spectra,
title = {{Spectra of Random Polynomial Matrices: the Petaloid Law}},
author = {Rikhav Shah and Edward Zeng},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.40232},
eprint = {2609.40232},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.40232},
}
RIS
TY - UNPB
TI - Spectra of Random Polynomial Matrices: the Petaloid Law
AU - Rikhav Shah
AU - Edward Zeng
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2609.40232
UR - https://arxiv.org/abs/2609.40232
AB - We study the distribution of the zeros of det P_N(z) where P_N(z) is a random monic polynomial matrix, i.e., P_N(z)=z^dI−∑_j=0^d−1A_j,Nz^j for possibly coupled random matrices A_j,N, scaled to have entrywise variance O(1/N). We provide general conditions under which this distribution almost-surely weakly converges as N→∞ to a deterministic measure, depending on just the variance and covariance of the entries of the A_j. This generalizes the circular, elliptic, and semicircle laws, which concern the special case of this question where d=1. Unlike those classical laws, these measures can combine nonuniform two-dimensional densities with singular components supported on curves, producing a variety of petal-shaped regions, inspiring our name ``the petaloid law''. We give explicit formulas for the limiting densities and supports. Under a Gaussianity assumption, we also show that there are almost surely no eigenvalues outside small neighborhoods of the limiting support.
ER -