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Convergence of the Laws of Non-Hermitian Sums of Projections

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Pith's one-line read Random matrices formed as P_n+iQ_n from two independently Haar-rotated two-valued Hermitian matrices converge to the Brown measure of the free limit.

desk verdict The two-atom convergence result is real and the Hermitization estimate is clean, but the constant-summand case has a genuine circular citation that needs a real proof. read the letter →

arxiv 2411.17159 v2 pith:PNHXC52I submitted 2024-11-26 math.OA

Max Sun Zhou This is my paper · ORCID
classification math.OA MSC 46L5460B20
keywords Brownmeasurefreeprobabilitynon-HermitianrandommatricesHermitizationminimumsingularvalueHaarunitaryrotationstwo-atomspectralmeasuresempiricaldistribution
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper establishes when the non-Hermitian random matrix $X_n = P_n + iQ_n$, with $P_n,Q_n$ independently Haar-rotated Hermitian matrices whose spectra have at most two points, has a deterministic limit. Under the assumption that the laws of $P_n$ and $Q_n$ converge to freely independent operators $p$ and $q$, the empirical spectral distribution of $X_n$ converges almost surely, in the vague topology, to the Brown measure of $X=p+iq$. The Brown measure is the natural candidate from free probability for the limiting eigenvalue law, and the result confirms it for this model. A converse is also proved: any vague limit of these empirical distributions is necessarily the Brown measure of such a free operator, so the two-atom model cannot produce other limit laws.

What carries the argument

The load-bearing mechanism is Hermitization: instead of studying the eigenvalues of $X_n$ directly, one proves that the logarithmic potentials $\frac12 \int_0^\infty \log x \, d\nu_{n,z}(x)$ converge to $\log \Delta(z-X)$, where $\nu_{n,z}$ is the spectral measure of $H_z(X_n)=(z-X_n)^*(z-X_n)$. The hard step is a deterministic minimum singular value estimate, Theorem 4.8: $\sigma_{\min}(z-X_n) \geq \frac{\operatorname{dist}(z,H_n\cap R_n)^2}{\|z-X_n\|}$. This estimate is proved using the algebra of two-valued Hermitian matrices: after centering, the square $\tilde X_n^2$ is normal with fixed real part, the eigenvalues of $X_n$ lie on the hyperbola-rectangle $H_n\cap R_n$, and the generalized eigenspaces have dimension at most two, which makes the singular-value calculation explicit on each invariant subspace.

What would settle it

Take a concrete two-atom example with fixed weights and positions, simulate $X_n$ at large $n$, and compare the empirical spectral distribution with the Brown measure of the free limit; a persistent mismatch, or a specific $z$ outside the limiting hyperbola-rectangle where $\sigma_{\min}(z-X_n)$ is asymptotically smaller than $\operatorname{dist}(z,H_n\cap R_n)^2/\|z-X_n\|$, would falsify the theorem.

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Extended reading notes

Core claim

The central claim is Theorem 5.1: for each $n$, let $P_n$ and $Q_n$ be independent Hermitian matrices obtained by Haar-unitary conjugation of deterministic two-atom Hermitian matrices, and set $X_n=P_n+iQ_n$. If the spectral laws of $P_n$ and $Q_n$ converge to those of freely independent Hermitian operators $p$ and $q$ in a tracial von Neumann algebra, then the empirical spectral distributions of $X_n$ converge almost surely in the vague topology to the Brown measure of $X=p+iq$. The Brown measure is the probability measure obtained by taking the distributional Laplacian of the Fuglede--Kadison determinant $\log \Delta(z-X)$; for normal operators it is the spectral measure, and here it is supported on the intersection of a hyperbola and a rectangle determined by the four atom positions. The paper also proves a converse, Theorem 5.2: if the empirical distributions converge in probability to a deterministic measure, then that measure is the Brown measure of a free operator of this form.

Load-bearing premise

The whole convergence argument rests on the deterministic lower bound for the smallest singular value of $z-X_n$ in Theorem 4.8, which is proved only when both $P_n$ and $Q_n$ have exactly two atom values; if that bound fails in any scaling regime, the logarithmic potentials do not converge and Theorem 5.1 does not follow.

Editorial extensions

If this is right

  • For any two-atom data with converging weights and positions, the eigenvalue distribution of $X_n$ has a deterministic limit, so large-$n$ simulations can be checked against explicit Brown-measure formulas.
  • The support of the limiting law is the hyperbola-rectangle $H\cap R$; the eigenvalues of $X_n$ cannot spread outside that curve in the limit.
  • If one of $p,q$ is a scalar, the theorem still delivers the corresponding Brown measure, covering degenerate two-atom-to-one-atom limits.
  • By the converse theorem, the map from the limiting laws of $P_n$ and $Q_n$ to the limiting eigenvalue law of $X_n$ is one-to-one, so no other deterministic vague limit is possible for this model.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A natural next step is to seek analogues of Theorem 4.8 for three or more atom values; the present proof uses that $\tilde X_n^2$ is normal, a property that already fails at three atoms, so a new mechanism would be needed.
  • Because the Brown measure determines the two atom laws uniquely, the empirical spectrum of a single large $X_n$ could in principle be used to recover the weights and positions of $P_n$ and $Q_n$; the paper does not discuss this statistical reading.
  • The bound's form, a squared distance to the support curve divided by the norm, suggests that quantitative convergence rates are controlled by how fast $H_n\cap R_n$ approaches $H\cap R$ in Hausdorff distance, a parameter-free geometric rate.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The manuscript studies the random matrix model X_n = P_n + i Q_n, where P_n and Q_n are independently Haar-rotated Hermitian matrices whose laws have at most two atoms. The main theorem (Theorem 5.1) asserts that if the laws of P_n and Q_n converge to the laws of freely independent Hermitian operators p and q, then the empirical spectral distribution of X_n converges almost surely in the vague topology to the Brown measure of X = p + iq. The proof uses the Hermitization method: asymptotic freeness gives convergence of the spectral measures of (z-X_n)^*(z-X_n), and an algebraic argument based on the special two-atom structure gives a lower bound for the minimum singular value of z-X_n in terms of the distance from z to the hyperbola-rectangle H_n ∩ R_n. A converse statement (Theorem 5.2) identifies possible vague limits as Brown measures of free limits. The nondegenerate two-atom case is treated in detail; the degenerate cases where p or q is constant are handled separately at the end of the proof of Theorem 5.1.

Significance. If the gap in the degenerate case is repaired, the paper provides a clean, explicit example of non-Hermitian random matrix convergence to a Brown measure, with a self-contained algebraic minimum-singular-value estimate that is likely to be useful beyond this model. The nondegenerate proof is coherent: Proposition 4.7 and Theorem 4.8 are internally consistent, and the reduction to logarithmic potentials via Proposition 2.7 is standard. The paper also gives a converse result, which is a welcome addition. The main weakness is that one load-bearing step in the proof of Theorem 5.1 relies on a companion-paper statement about the Brown measure of the limiting operator rather than on an argument about the finite empirical measures; this is a circularity that must be resolved. The Hausdorff-convergence step is also asserted through an unstated corollary of the companion paper, though this is likely repairable.

major comments (2)
  1. [Section 5, proof of Theorem 5.1, exactly-one-constant case] The final paragraph of the proof asserts that when both atoms of P_n lie near the scalar p, 'the branches of H_n ∩ R_n ... have the appropriate measures µ_{Q_n}({β_n}) = b_n and µ_{Q_n}({β'_n}) = 1-b_n', citing [13, Proposition 4.7]. That proposition concerns the Brown measure of the limiting operator p+iq, not the empirical spectral measure of the finite random matrix P_n+iQ_n. Using it here is circular, because Theorem 5.1 is precisely the statement that the empirical measures converge to that Brown measure. No trace computation, Hermitization estimate, or perturbation argument is supplied for the finite-n branch weights. This case is not a negligible afterthought: it covers all models with a scalar free summand, and without it Theorem 5.1 is not proven as stated.
  2. [Section 5, Eq. (86)-(87)] The proof that sup_{w ∈ H_n ∩ R_n} dist(w, H ∩ R) → 0 is asserted via [13, Corollary 4.5], but the corollary is not stated and no estimate is given. This step is load-bearing for the lower bound dist(z, H_n ∩ R_n) ≥ δ for z outside H ∩ R, which in turn is needed to apply Theorem 4.8 and to keep the logarithmic integrals away from the singularity at 0. Since the parameters α_n, α'_n, β_n, β'_n converge, a direct parameterization of the two hyperbola-rectangle intersections should suffice, but as written the argument depends on an unstated external result.
minor comments (4)
  1. [Section 4, Proposition 4.3] In the proof, 'Since Re( ˜Xn) is a constant' should read 'Since Re( ˜Xn²) is a constant', and the conclusion should be that ˜Xn² is normal (and diagonalizable), not that ˜Xn is normal.
  2. [Section 5, after Theorem 4.8] If σ_min(z - X_n) > δ, then the spectrum of H_z(X_n) = (z-X_n)^*(z-X_n) is contained in [δ², M], not [δ, M] as written. The subsequent support argument should use δ² (or relabel δ).
  3. [Section 5, proof of Theorem 5.2] In the tightness argument for Q_n, the text says that certain events hold 'with positive probability'; since convergence in probability gives probability tending to 1, this wording should be strengthened, and the intended conclusion follows cleanly.
  4. [Section 5, proof of Theorem 5.2] The sentence 'This implies that µPn → δ0. For this to happen, the weight of the atom at 1, a_n, has to tend to 0' is confusing: after rescaling, the atom at 1 refers to the scaled variable P'_n, not the original P_n. The notation should be adjusted to avoid this ambiguity.

Circularity Check

1 steps flagged · score 6.0 of 10

Degenerate case of Theorem 5.1 imports branch-weight conclusion from the author's companion paper [13], which computes the limiting Brown measure, not the finite-n empirical measure.

  1. self citation load bearing [Section 5, proof of Theorem 5.1, final paragraph (case where exactly one of p and q is constant).]
    "For the case that both atoms of Pn are in a small neighborhood of p, the branches of Hn ∩ Rn are in small neighborhoods of {p + iβ, p+ iβ′}, and from [13, Proposition 4.7] these branches have the appropriate measures µQn ({βn}) = bn and µQn ({β′n}) = 1 − bn so that µn tends towards µ = bδβ + (1 − b)δβ′."

    The theorem's conclusion for this case is that the empirical measure µn of Pn + iQn converges to the Brown measure of p + iq. The cited [13, Proposition 4.7] is a computation of that Brown measure for the limiting operator, not a statement about the finite-n empirical spectral measure µn. Assigning the two branches of Hn ∩ Rn the masses bn and 1 − bn therefore assumes exactly the convergence being proved: no trace calculation or Hermitization bound in this paper establishes the finite-n branch weights. The self-citation carries the load for the degenerate case, so Theorem 5.1 as stated is not derived for this case without importing its own conclusion.

full rationale

The main two-atom case is largely self-contained: Proposition 3.1 and Corollary 3.2 use standard asymptotic freeness, and Theorem 4.8 derives the minimum singular value bound from the algebraic identities of Section 4 rather than from the target convergence. The use of [13] to identify the candidate Brown measure and to parameterize the hyperbola and rectangle is legitimate, since those are independent computations of the limit object. However, the final case of Theorem 5.1, where exactly one of p and q is constant, invokes [13, Proposition 4.7] to assign empirical branch masses for finite n. Since [13] computes the Brown measure of the limiting operator, that assertion is circular: it assumes the very convergence the theorem must prove. This is a genuine load-bearing gap, not merely a citation of an unrelated known result. Score 6 reflects that the central two-atom proof is independent, but the theorem as stated also covers all scalar-plus-two-atom models, and that case is not derived without circular reliance on the companion paper.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The model's constants α, α', β, β', a, b are parameters of the input laws, not fitted to the result. No extra free parameters or invented objects appear; the only external content is the companion paper's Brown measure formulas.

assumptions (4)
  • standard math Asymptotic freeness of independent Haar unitaries from deterministic matrices (Theorem 5.4.10 of Anderson-Guionnet-Zeitouni).
    Used in Proposition 3.1 to get joint convergence of P_n and Q_n to free p and q.
  • domain assumption Brown measure formula and unicity for X = p + iq from [13], including Corollary 2.4 and Proposition 4.7.
    Section 2.2 and the proof of Theorem 5.1 use the companion paper's explicit Brown measure; it is not rederived here.
  • standard math Hermitization theorem (Proposition 2.7, cited from Tao, Topics in Random Matrix Theory, Theorem 2.8.3) that convergence of logarithmic potentials implies vague convergence of measures.
    Section 2.3 relies on this to pass from logarithmic potential convergence to ESD convergence.
  • standard math Standard properties of the Fuglede-Kadison determinant and Brown measure from Brown, Mingo-Speicher, and Haagerup-Schultz.
    Section 2.1 uses these to define the Brown measure and its Riesz representation.

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Pith. "Pith review of Convergence of the Laws of Non-Hermitian Sums of Projections." pith.science (2026). https://pith.science/paper/PNHXC52I

@misc{pith2026241117159,
  author       = {Pith},
  title        = {Pith review of: Convergence of the Laws of Non-Hermitian Sums of Projections},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PNHXC52I}},
  note         = {Machine review of arXiv:2411.17159}
}
abstract

We consider the random matrix model $X_n = P_n + i Q_n$, where $P_n$ and $Q_n$ are independently Haar-unitary rotated Hermitian matrices with at most $2$ atoms in their spectra. Let $(M, \tau)$ be a tracial von Neumann algebra and let $p, q \in (M, \tau)$, where $p$ and $q$ are Hermitian and freely independent. Our main result is the following convergence result: if the law of $P_n$ converges to the law of $p$ and the law of $Q_n$ converges to the law of $q$, then the empirical spectral distributions of the $X_n$ converges to the Brown measure of $X = p + i q$. To prove this, we use the Hermitization technique introduced by Girko, along with the algebraic properties of projections to prove the key estimate. We also prove a converse statement by using the properties of the Brown measure of $X$.

Figures

Figures reproduced from arXiv: 2411.17159 by the authors.

Figure 1
Figure 1. Xn = Pn + iQn µPn ≈ (5/8)δ0 + (3/8)δ1 µQn ≈ (7/8)δ0 + (1/8)δ4/5 We also prove a converse result in Theorem 5.2 that the Brown measure of operators of the form X = p + iq are the only possible limits for the empirical spectral distributions of Xn: Theorem 5.2. Consider the random matrix model Xn = Pn +iQn, where Pn, Qn ∈ Mn(C) are independently Haar-rotated Hermitian matrices with distributions with at most 2 atoms. … view at source ↗

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Quaternionic Green's Function and the Brown Measure of Atomic Operators

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    For free atomic Hermitian operators p and q, the boundary heuristic for the Brown measure of X = p + iq implies the boundary is an algebraic curve, and the paper provides an explicit resultant-based algorithm to produ...

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