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arxiv: 1906.10674 · v1 · pith:2PYNECBRnew · submitted 2019-06-25 · 🧮 math.PR

Outlier eigenvalues for non-Hermitian polynomials in independent i.i.d. matrices and deterministic matrices

Serban Belinschi , Charles Bordenave , Mireille Capitaine , Guillaume C\'ebron This is my paper

Pith reviewed 2026-05-25 15:55 UTC · model grok-4.3

classification 🧮 math.PR
keywords outlier eigenvaluesnon-Hermitian random matricesnoncommutative polynomialsfree probabilitycircular systems*-distribution convergence
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The pith

A sufficient condition ensures outlier eigenvalues of P(Y,A) match those of P(0,A) asymptotically.

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

The paper considers square random matrices formed by a noncommutative polynomial P applied to a tuple Y of independent i.i.d. centered matrices with variance 1/N and a tuple A of deterministic matrices. The deterministic matrices A converge in *-distribution to a tuple a in a C*-probability space, and the analysis focuses on eigenvalues of P(Y,A) that lie outside the spectrum of P(c,a) where c is a circular system free from a. Under a sufficient condition, these outlier eigenvalues coincide asymptotically with the eigenvalues of P(0,A). A reader would care because the result isolates the contribution of the deterministic part from the random circular noise in the limiting spectrum of polynomial matrix models.

Core claim

We provide a sufficient condition to guarantee that the eigenvalues of P(Y,A) outside the spectrum of P(c,a) coincide asymptotically with those of P(0,A).

What carries the argument

The sufficient condition on the *-distribution convergence of the deterministic tuple A, which separates the outlier eigenvalues from the circular bulk spectrum generated by the free circular system c.

If this is right

  • The outlier part of the spectrum is fully determined by the deterministic matrices evaluated at zero random input.
  • The result holds for arbitrary noncommutative polynomials P.
  • The circular system c accounts for all bulk eigenvalues contributed by the random i.i.d. part Y.

Where Pith is reading between the lines

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

  • The same separation between outliers and circular bulk may hold for entry distributions other than i.i.d. centered with variance 1/N.
  • The condition could be used to study stability of large non-Hermitian systems that combine fixed deterministic blocks with random noise.
  • Numerical checks on moderate-sized matrices could test whether the asymptotic coincidence appears already at finite N.

Load-bearing premise

The deterministic matrices A converge in *-distribution toward a tuple a in a C*-probability space, with c a circular system free from a.

What would settle it

For large N satisfying the sufficient condition, compute the eigenvalues of P(Y,A) and check whether any outside the P(c,a) spectrum fail to match those of P(0,A); a mismatch would falsify the claim.

Figures

Figures reproduced from arXiv: 1906.10674 by Charles Bordenave, Guillaume C\'ebron, Mireille Capitaine, Serban Belinschi.

Figure 1
Figure 1. Figure 1: In black, the eigenvalues of P1  X (1) √N N , X (2) √N N , X (3) √N N , AN  for N = 1000, and in red, the outliers 2.5 and 2i − 0.5 of P1(0N , 0N , 0N , AN ). 2.2 Example 2 We consider the matrix MN = P2 [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: In black, the eigenvalues of P2  X (1) √N N , X (2) √N N , X (3) √N N , A(1) N , A(2) N  for N = 1000, and in red, the limiting outliers 2.125 and −2.6 of P1(0, 0, 0, A(1) N , A(2) N ). The matrix MN converges in ∗-distribution to the elliptic variable 1 2 (c+s), where c is a circular variable and s a semicircular variable free from c. The empirical spectral measure of MN converges to the Brown measure o… view at source ↗
Figure 3
Figure 3. Figure 3: In black, the eigenvalues of P3  X (1) √N N , X (2) √N N , X (3) √N N , A(1) N , A(2) N  for N = 1000, and in red, the outliers 2.5 and −1 + 2i of P3(0N , 0N , 0N , A(1) N , A(2) N ). 2.4 Example 4 We consider the matrix MN = P4 [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: In black, the eigenvalues of P4  X (1) √N N , X (2) √N N , X (3) √N N , AN  for N = 1000, and in red, the outliers −2 + 2.4i and −2 − 2.4i of P4(0N , 0N , 0N , AN ). pected (but not proved) that the empirical spectral measure of MN converges to the Brown measure of (c1 + 3) (c2 + 2) (c3 + 2) /5 − 2. The spectrum of (c1 + 3) (c2 + 2) (c3 + 2) /5 − 2 is included in the set (B(0, 1) + 3) (B(0, 1) + 2) (B(0,… view at source ↗
read the original abstract

We consider a square random matrix of size $N$ of the form $P(Y,A)$ where $P$ is a noncommutative polynomial, $A$ is a tuple of deterministic matrices converging in $\ast$-distribution, when $N$ goes to infinity, towards a tuple $a$ in some $\mathcal{C}^*$-probability space and $Y$ is a tuple of independent matrices with i.i.d. centered entries with variance $1/N$. We investigate the eigenvalues of $P(Y,A)$ outside the spectrum of $P(c,a)$ where $c$ is a circular system which is free from $a$. We provide a sufficient condition to guarantee that these eigenvalues coincide asymptotically with those of $P(0,A)$.

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 / 1 minor

Summary. The manuscript considers square N x N random matrices of the form P(Y,A), where P is a noncommutative polynomial, A is a tuple of deterministic matrices that converge in *-distribution to a tuple a in a C*-probability space, and Y is a tuple of independent matrices with i.i.d. centered entries of variance 1/N. It studies the eigenvalues of P(Y,A) lying outside the spectrum of P(c,a), where c is a circular system free from a, and provides a sufficient condition guaranteeing that these outlier eigenvalues asymptotically coincide with those of the deterministic matrix P(0,A).

Significance. If the sufficient condition is correctly identified and the supporting arguments hold, the result extends existing free-probability techniques for outlier eigenvalues from linear or simple non-Hermitian cases to general noncommutative polynomials involving both random i.i.d. and deterministic components. This is a natural and potentially useful contribution to the spectral theory of non-Hermitian random matrices.

minor comments (1)
  1. The abstract states the existence of a sufficient condition but does not indicate its form or the key technical steps used to establish it; adding a brief indication of the condition (e.g., a reference to the relevant theorem number) would improve readability without altering the technical content.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading and positive assessment of the manuscript. The recommendation of minor revision is noted. However, the report contains no specific major comments to address.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper states a sufficient condition for outlier eigenvalues of P(Y,A) to asymptotically match those of P(0,A), under the standard assumption that deterministic matrices A converge in *-distribution to a tuple a in a C*-probability space with c a free circular system. This setup uses established free-probability objects and convergence notions that are not defined in terms of the paper's own results or fitted quantities. No self-definitional reductions, fitted inputs presented as predictions, or load-bearing self-citation chains appear in the abstract or described claims. The central statement remains an independent sufficient-condition result within the existing framework.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only abstract available; no explicit free parameters, axioms, or invented entities can be extracted beyond the standard setup of free probability and *-distribution convergence mentioned in the abstract.

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