{"id":"ae5173d0-a99f-4ad5-842b-1626057e2713","arxiv_id":"2411.17159","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For random matrices X_n = P_n + iQ_n with P_n and Q_n independently Haar-rotated Hermitian two-atom matrices, the empirical spectral distribution converges almost surely to the Brown measure of p + iq whenever the laws converge.","lead":"This paper proves that eigenvalue distributions of a class of random non-Hermitian matrices, built from two independently rotated two-point matrices, converge to the Brown measure of a free operator. It also proves a converse, showing these Brown measures are the only possible vague limits for this model.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Scalar-case branch-weight assertion in Theorem 5.1 rests on a circular citation; the two-atom Hermitization proof itself appears sound.","rationale":"The reader's flagged Hausdorff convergence is actually minor: since α_n, β_n and the weights converge, the sets H_n∩R_n are uniformly bounded, and any limit point of w_n∈H_n∩R_n satisfies the limiting equations, so the asserted sup-distance convergence follows by compactness. The minimum singular value estimate also checks out; Proposition 4.7 is internally consistent, and the symmetry of the hyperbola-rectangle justifies the distance comparison in Theorem 4.8. The less visible but more serious gap is the scalar-case argument: the final paragraph of Theorem 5.1 assigns finite-n branch masses using a companion-paper proposition about the limiting Brown measure, which is circular in context. This does not undermine the central two-atom Hermitization mechanism, but it leaves a real hole in the theorem as stated. The reader's conditional verdict remains appropriate; the requested fixes should be augmented with a direct proof of the scalar-case branch weights, ideally by running the same Hermitization argument outside the two limiting atoms.","tokens_in":18429,"tokens_out":29176,"duration_ms":272808,"concrete_test":"Fill the gap analytically: for a scalar case α_n=p+ε_n, α'_n=p−ε_n with ε_n→0 and Q_n two-atom, compute the limiting logarithmic potentials using the Section 4 bound for z outside {p+iβ,p+iβ'} and Corollary 3.2. If the potentials converge to the atomic Brown measure, the branch-weight assertion can be proven and the circular citation removed. Alternatively, a numerical check at n=1000 with ε_n=10^{-2} and β−β'=1 should show the ESD mass in small disjoint neighborhoods of the two atoms tracking b_n and 1−b_n; a mismatch would falsify the assertion.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The main two-atom proof is coherent: Proposition 4.7 is valid, and in Theorem 4.8 the distance comparison is salvaged by the symmetry of H_n∩R_n (both square roots of a nonzero ρ lie on the hyperbola-rectangle), so the minimum singular value estimate is not the real risk. The load-bearing gap is the degenerate case at the end of the proof of Theorem 5.1, where exactly one of p,q is constant. The text asserts that when both atoms of P_n lie near p, the two branches of H_n∩R_n carry spectral mass equal to b_n and 1−b_n, and cites [13, Proposition 4.7] for this. 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 calculation or Hermitization argument is supplied for the branch weights. This case is not a negligible afterthought: it includes all models with a scalar free summand, and without it Theorem 5.1 is not proven as stated.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":18600,"tokens_out":20495,"duration_ms":192454,"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":[{"comment":"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.","section":"Section 5, proof of Theorem 5.1, exactly-one-constant case"},{"comment":"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.","section":"Section 5, Eq. (86)-(87)"}],"minor_comments":[{"comment":"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.","section":"Section 4, Proposition 4.3"},{"comment":"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).","section":"Section 5, after Theorem 4.8"},{"comment":"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.","section":"Section 5, proof of Theorem 5.2"},{"comment":"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.","section":"Section 5, proof of Theorem 5.2"}],"recommendation":"major_revision","confidential_remarks":"The manuscript relies heavily on results from the companion preprint [13], including the Brown measure formula, the injectivity of the map (µ_p, µ_q) ↦ µ, the parameterization of H ∩ R, and Proposition 4.7. The circular use of [13, Proposition 4.7] in the scalar case should be fixed within this manuscript, and [13, Corollary 4.5] should be stated explicitly or proved. The editor may wish to verify that [13] is available and correct, as the present paper's completeness depends on it."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Zhou proves that for independent Haar-rotated Hermitian matrices with two-point spectra, the empirical spectral distribution of P_n + iQ_n converges to the Brown measure of the free limit p+iq, and a converse. The main theorem is new and the machinery is mostly sound. The minimum singular value estimate (Theorem 4.8) is the real contribution: the reduction to two-dimensional invariant subspaces and the hyperbola-rectangle geometry is clean and convincing. The generic two-atom case is handled properly: asymptotic freeness gives spectral measure convergence, the singular value bound gives the log-potential convergence, and the support geometry rules out the exceptional set.\n\nThe soft spots. The proof of the degenerate case where exactly one of p,q is constant is not adequate. The text asserts that when both atoms of P_n are near p, the branches of H_n∩R_n carry mass b_n and 1−b_n, and cites [13, Proposition 4.7] for it. That proposition is about the Brown measure of the limiting operator, not about the finite-matrix empirical measure. Using it here is circular: Theorem 5.1 is exactly the convergence of the empirical measures to that Brown measure. No trace computation for the branch weights is supplied. This case includes all models with a scalar free summand, so as written Theorem 5.1 is not fully proven. I think it can be fixed — a direct computation of τ(χ_{P_n}(near p) ∧ ...) or a limiting argument from the two-atom case — but it needs to be in the paper.\n\nAlso, the Hausdorff convergence H_n∩R_n → H∩R is asserted via [13, Cor 4.5] without a proof or estimate. That is probably fine because the parameterization makes it obvious, but it should be one line. There's a typo in Proposition 4.3 (\"Re(˜X_n) is a constant\" should be Re(˜X_n^2)). Minor.\n\nThe reliance on the companion paper [13] for the Brown measure formulas and unicity is acceptable; those are independent computations, and the unicity result (Corollary 2.4) is what makes the converse work.\n\nVerdict: worth refereeing. The generic case is solid and novel, the singular value estimate is a genuinely new ingredient, and the converse is a nice addition. The referee should focus on the constant-summand case and ask for a real proof there. If the author supplies it, this is a good paper for the field.","headline":"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.","tokens_in":19151,"tokens_out":2315,"would_cite":true,"duration_ms":19751,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["46L54","60B20"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["Brown measure","free probability","non-Hermitian random matrices","Hermitization","minimum singular value","Haar unitary rotations","two-atom spectral measures","empirical spectral distribution"],"falsifier":"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.","tokens_in":18178,"feed_emoji":"📊","tokens_out":9605,"duration_ms":79302,"temperature":0.7,"pith_summary":"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.","feed_headline":"Two-valued Haar-rotated matrices converge to Brown measure","feed_subtitle":"A non-Hermitian model from independently rotated two-point Hermitian matrices has a provable limiting eigenvalue law.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Computes the Brown measure of X=p+iq, gives the hyperbola-rectangle geometry, the Hausdorff convergence of H_n∩R_n, and the injectivity used in the converse.","marker":"[13]"},{"why":"Introduces the Hermitization method that reduces convergence of non-Hermitian empirical spectral distributions to convergence of logarithmic potentials.","marker":"[4]"},{"why":"Supplies the criterion that vague convergence of measures follows from almost-everywhere convergence of their logarithmic potentials.","marker":"[10]"},{"why":"Provides the asymptotic freeness theorem for independent Haar unitaries that identifies the limit of the joint law of P_n and Q_n.","marker":"[1]"},{"why":"Defines the Brown measure as the distributional Laplacian of the Fuglede–Kadison determinant and establishes its basic properties.","marker":"[2]"},{"why":"Defines the Fuglede–Kadison determinant used in the logarithmic potential of the limit operator.","marker":"[3]"}],"fun_headline_variants":["Haar-rotated two-point Hermitians converge to Brown measure","Two-valued Haar rotations: eigenvalue law converges to Brown measure","Non-Hermitian sums of projections have Brown measure limits","Convergence of laws for non-Hermitian sums of projections","Brown measure limit for rotated two-atom projections"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Haar-rotated two-point Hermitians converge to Brown measure","Two-valued Haar rotations: eigenvalue law converges to Brown measure","Non-Hermitian sums of projections have Brown measure limits","Convergence of laws for non-Hermitian sums of projections","Brown measure limit for rotated two-atom projections"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00065,"raw_usage":{"total_tokens":2981,"prompt_tokens":943,"completion_tokens":2038,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":559,"completion_tokens_details":{"reasoning_tokens":1954}},"tokens_in":559,"tokens_out":2038,"duration_ms":13961,"temperature":1.0,"reasoning_tokens":1954,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T12:27:44.856346+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"The circular law","cited_arxiv_id":null,"evidence_quote":"Introduces the Hermitization method that reduces convergence of non-Hermitian empirical spectral distributions to convergence of logarithmic potentials."},{"cited_title":"Graduate Studies in Mathe- matics","cited_arxiv_id":null,"evidence_quote":"Supplies the criterion that vague convergence of measures follows from almost-everywhere convergence of their logarithmic potentials."},{"cited_title":"Anderson, Alice Guionnet, and Ofer Zeitouni.An Introduction to Random Matrices","cited_arxiv_id":null,"evidence_quote":"Provides the asymptotic freeness theorem for independent Haar unitaries that identifies the limit of the joint law of P_n and Q_n."}],"review_version":1}