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arxiv: 2606.01664 · v1 · pith:INWSZ7WTnew · submitted 2026-06-01 · 🧮 math.PR

Brown measure convergence for the spectrum of polynomials in Ginibre matrices

Yi Han This is my paper

Pith reviewed 2026-06-28 13:17 UTC · model grok-4.3

classification 🧮 math.PR
keywords Brown measureGinibre matricesempirical spectral distributionfree probabilityrandom matricesleast singular valuenon-Hermitian matrices
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The pith

The spectrum of any fixed polynomial in Ginibre matrices converges in distribution to the Brown measure of the same polynomial in free circular elements.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that for any multivariate polynomial p of any degree, the eigenvalues of the matrix p applied to n independent Ginibre matrices converge weakly to the Brown measure obtained by evaluating the same polynomial at free circular variables. Earlier work had established this only for polynomials of degree at most two; the extension to arbitrary degree rests on a new lower bound for the smallest singular value of the shifted matrix P^N minus z for almost every complex z. The same convergence is proved when the matrix entries are drawn from a wider class of distributions that have mean zero, variance one, bounded density, and finite moments of all orders.

Core claim

The empirical spectral distribution of P^N = p(X_1^N, …, X_n^N) converges weakly to the Brown measure of p evaluated at free circular variables, for every fixed non-commuting polynomial p of arbitrary degree.

What carries the argument

A least singular value lower bound for P^N − z (for almost all z), obtained by reducing to a corresponding bound for tensorized Ginibre matrices of finite type with a deterministic shift.

If this is right

  • The convergence result is no longer restricted to quadratic or lower-degree polynomials.
  • The same limiting Brown measure appears when the Ginibre entries are replaced by i.i.d. random variables with mean zero, variance one, bounded density, and all moments finite.
  • A new least-singular-value estimate is available for a broad family of tensorized Ginibre matrices shifted by a deterministic matrix.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The singular-value technique may extend to other non-Hermitian random-matrix models whose entries satisfy comparable moment and density conditions.
  • The result supplies a concrete way to compute limiting spectra of high-degree non-commutative polynomials without having to diagonalize large random matrices directly.

Load-bearing premise

A uniform lower bound on the smallest singular value of the shifted matrix P^N − z holds for almost every complex number z.

What would settle it

For some polynomial p of degree three or higher, compute the empirical spectral distribution of P^N for large N and observe that its weak limit differs from the Brown measure of p at free circular variables.

read the original abstract

Fix a multivariate polynomial $\mathfrak{p}$ in $n$ non-commuting variables of arbitrary degree, and consider $n$ independent $N\times N$ complex Ginibre matrices $X_1^N,\cdots,X_n^N$. We prove that the empirical spectral distribution of $P^N=\mathfrak{p}(X_1^N,\cdots,X_n^N)$ converges as $N$ tends to infinity to the so-called Brown measure of $\mathfrak{p}$ evaluated at free circular variables. For polynomials of degree at most 2, the convergence was proven by Cook, Guionnet, and Husson \cite{cook2022spectrum}, and we prove that the convergence in fact holds for polynomials $\mathfrak{p}$ of any degree. The main step in the proof is a least singular value lower bound for $P^N-z$ for almost all complex shifts $z$, and we prove this via a least singular value lower bound for a wide class of tensorized Ginibre matrices of finite type with a deterministic shift, which is of independent interest. We further show that the Brown measure convergence holds beyond Gaussians: the same convergence holds when the entry law has mean 0, variance 1, bounded density on $\mathbb{C}$ and finite moments of all orders.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript proves that for any fixed multivariate polynomial 𝔭 of arbitrary degree in n non-commuting variables, the empirical spectral distribution of P^N = 𝔭(X_1^N, …, X_n^N) converges to the Brown measure of 𝔭 evaluated at free circular elements, where the X_i^N are independent complex Ginibre matrices. This extends the degree-at-most-2 result of Cook–Guionnet–Husson by establishing a new least-singular-value lower bound for P^N − z (almost every z) via a corresponding bound for tensorized Ginibre matrices of finite type with deterministic shift; the same convergence is shown to hold when the entries are i.i.d. with mean zero, unit variance, bounded density on ℂ, and all moments finite.

Significance. The result substantially enlarges the class of non-Hermitian random-matrix models whose limiting spectral measures are known explicitly via free probability. The least-singular-value bound for tensorized Ginibres is of independent interest and supplies a technical tool that may be reusable in other contexts. The non-Gaussian extension further increases the robustness of the statement.

minor comments (2)
  1. [Abstract] Abstract, line 3: the phrase “tensorized Ginibre matrices of finite type” is introduced without a forward reference; a brief parenthetical definition or pointer to the relevant section would improve immediate readability.
  2. The citation \cite{cook2022spectrum} appears in the abstract; confirm that the bibliography entry is complete and that the year and title match the published version.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, for highlighting its significance in extending Brown measure convergence results to arbitrary-degree polynomials in Ginibre matrices, and for recommending acceptance.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's central result extends the known degree-2 convergence of Cook-Guionnet-Husson by proving an independent least singular value lower bound for tensorized Ginibre matrices with deterministic shift; this bound is invoked to upgrade the prior result to arbitrary fixed degree without any redefinition of the Brown measure in terms of the empirical spectral distribution. The Brown measure itself is a standard object from free probability theory applied to free circular elements, not constructed or fitted from the paper's convergence statement. No self-definitional steps, fitted-input predictions, load-bearing self-citations, or ansatz smuggling occur; the argument remains self-contained against external benchmarks in free probability and random matrix theory.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit list of free parameters or invented entities; the least-singular-value bound is presented as a new derived estimate rather than an axiom.

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