A mathematical preprint reports that the zeros of a class of random polynomials converge, as matrix dimensions grow, to a semicircle distribution on the imaginary axis. The result concerns permanental roots—zeros of a characteristic polynomial expanded from principal submatrices—and is established almost surely for sequences drawn from the Gaussian orthogonal ensemble, or GOE, and Gaussian unitary ensemble, or GUE.
At a general-reader level, the result says the roots have a predictable large-scale shape. That shape is the standard Wigner semicircle law from -2 to 2 turned through π/2, so it lies along the imaginary axis. The theorem describes the roots in aggregate; it does not establish quantitative confinement of individual roots to the imaginary axis and limiting interval.
A theorem about growing matrices
The analysis follows a sequence of random matrices rather than one fixed dimension. For each N ≥ 1, the matrices are defined on a common probability space, with β = 1 for GOE and β = 2 for GUE.
The main statement uses weak convergence, a mathematical way of saying that the normalized counting measure of the roots approaches the target distribution as N grows. The result can also be tested through any bounded continuous function: average such a function over the roots, and that average converges almost surely to the corresponding integral along the rotated semicircle from -2 to 2.
The authors say the theorem proves Fyodorov’s conjecture about the global rotated semicircle law for GOE and GUE permanental roots. The supplied document is an arXiv version 1 preprint dated 26 August 2026.
Inside the proof
To build the polynomial, the proof expands the permanent characteristic polynomial as a sum over principal submatrices, producing a monic polynomial. For a Hermitian matrix, the polynomial has real coefficients.
The key ensemble calculation uses two-point identities for GOE and GUE. They reduce expectations evaluated at ix and -ix to two-dimensional Gaussian integrals involving the coordinate difference raised to powers 2 and 4.
That calculation is applied to the rotated monic polynomial R_N(z) = i^(−N)P_N(iz). Its expected Gaussian weighted L2 norm—a weighted squared-size measure—is exactly κβ,N h_N. The proof identifies κ1,N as (N + 2)/2 for GOE and κ2,N as N + 1 for GUE, and bounds κβ,N by 9N² for N ≥ 1.
The final step uses a weighted-polynomial criterion: under the stated expected L2 bound, zeros of a sequence of random monic polynomials converge almost surely to the semicircle law. With A = 9 and m = 2, the criterion gives almost-sure semicircle convergence for the zeros of the rotated polynomial.
What remains unresolved
The proof uses identities special to Gaussian ensembles, and extension to non-Gaussian real symmetric and complex Hermitian Wigner matrices is left as a universality problem.
The result is global and asymptotic. The supplied analysis identifies quantitative confinement of roots to the imaginary axis and to the limiting interval as an open question. In practical terms, it does not supply a finite-dimension bound on how far individual roots can stray.
Paper data and sources
Original title: Rotated semicircle laws for permanental roots of Gaussian random matrices
Authors: Renjie Feng, Dong Yao
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text