On the Brown measure of $x + i y$, with $x,y$ selfadjoint and $y$ free Poisson

Alexandru Nica; Franz Lehner; Kamil Szpojankowski; Ping Zhong

REVIEW 3 major objections 4 minor 23 references

On the Brown measure of $x + i y$, with $x,y$ selfadjoint and $y$ free Poisson

T0 review · 3 major / 4 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read A subordination left inverse yields an explicit density formula for the Brown measure of x + i y with y free Poisson.

desk verdict A genuinely new Brown-measure computation for x+iy with y free Poisson; the main theorem is right in substance, but Proposition 5.2 has sign typos that make the printed proof hard to trust, and the abstract/Theorem 1.8 have a variable typo. read the letter →

arxiv 2512.23528 v2 pith:PMAVD2BD submitted 2025-12-29 math.OA math-phmath.MPmath.PR

Franz Lehner , Alexandru Nica , Kamil Szpojankowski , Ping Zhong This is my paper · ORCID

classification math.OAmath-phmath.MPmath.PR MSC 46L5447A10

keywords BrownmeasurefreePoissonsubordinationprobabilityoperator-valuedCauchytransformnon-Hermitianrandommatricesspectral

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

The paper establishes that, when y is a free Poisson element freely independent of a selfadjoint x, the Brown measure of x+iy has its support inside the closure of a region M, and on M its density is given by an explicit formula in terms of the inverse of a change of variables h. This matters because explicit Brown-measure computations are rare: most known cases rely on special structure, whereas here the formula is systematically derived from the matrix-valued subordination function of the hermitization. The key move is to use the explicit left inverse H of the subordination function to construct h, so the density becomes computable from the derivatives of h^{-1}. A fully worked example (x with symmetric Bernoulli law, p=1) produces a concrete algebraic support and density matching random-matrix simulations.

What carries the argument

The engine is the matrix-valued subordination function Ω of the hermitization of x+iy, together with its explicit left inverse H(B)=B+p(J−G_X(B))^{-1} obtained from the rational R-transform of the free Poisson element. H is used to define the domain D (points λ where a certain unimodular-type limit is negative), the positive function δ_0(λ) satisfying H_11(λ, iδ_0)=0, and the reparametrization h(λ)=H_12(λ, iδ_0(λ)). The identity H(Ω(B))=B and the strict positivity of a 3x3 Jacobian (Proposition 5.2) make h a diffeomorphism and allow the logarithmic potential L_{x+iy} to be extended real-analytically, from which the density is read off by applying the Laplacian.

What would settle it

For a concrete µ_x (say x with symmetric Bernoulli law), compute det J Ĥ(α,β,δ_0(α+iβ)) from Proposition 5.2 at a dense grid of points of D; finding any point where it is zero or negative would falsify the claim. A softer check: verify numerically that the density formula integrates to the total Brown mass over M.

Watch

Extended reading notes

Core claim

The central result is Theorem 1.8 (restated as Theorem 6.4): for s+it in the open set M=h(D), the absolutely continuous density of the Brown measure of x+iy is f(s,t)=1/(4π)[(2/t)(∂α/∂s+∂β/∂t)−2/t−2β/t^2], where α+iβ=h^{-1}(s+it). Here D and M are open subsets of C built from the left inverse H(B)=B+p(J−G_X(B))^{-1} of the subordination function Ω of the hermitization of x+iy, and h(λ) is the off-diagonal entry of H at the unique δ_0(λ)>0 where the diagonal entry vanishes. The support of the Brown measure is contained in cl(M), and under Assumption 1.6 (spec(x)⊆cl(D), automatic when p>1) the region outside cl(M) carries no mass. In the Bernoulli example the support coincides with the spectru

Load-bearing premise

The whole construction rests on the unproved assertion in Proposition 5.2 that the 3x3 Jacobian of the reparametrized map is strictly positive inside D; if that determinant can vanish for some distribution of x, the diffeomorphism h and the density formula collapse.

Editorial extensions

If this is right

If the main theorem is correct, the Brown measure of x+iy is fully described by an explicit, checkable formula whenever one can invert h — no limiting random-matrix heuristics needed.
For p>1, the computation is unconditional in the sense that Assumption 1.6 is automatic; for p≤1 the supplementary conditions on µ_x are explicit and testable.
The Bernoulli example shows the method produces an algebraic boundary for the support and a closed-form density whose singular behavior at ±1 is visible in simulations.
The same subordination-left-inverse strategy is proposed as a general methodology that may extend to other distributions of y, including the semicircular case.

Reading between the lines

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

A direct numerical test of the density formula would be to compute h^{-1} for a grid in M and compare the resulting f with eigenvalue histograms of large random-matrix models; the paper's own Bernoulli figure does this for one case, but other µ_x are untested.
The identity between the Brown-measure support and the spectrum, verified in the Bernoulli example, is suggested as a general phenomenon; the paper only proves support containment, so checking spectrum-vs-support agreement for other µ_x would sharpen the statement.
The method's reliance on an explicit left inverse H suggests that other distributions of y with a tractable rational R-transform (for instance, certain elliptic or compound free Poisson laws) may admit analogous density formulae.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Here's my take. The paper does something new: it gives an explicit absolutely continuous density for the Brown measure of x+iy when y is free Poisson, with a reparametrization h constructed via the left inverse H of the subordination function. That's not in the existing literature — the semicircular and elliptic cases were known. The worked Bernoulli/p=1 example is fully explicit and matches the matrix model. The construction of h from H is a genuinely useful device; it plausibly generalizes to other y.

The main formula is derived, not fitted. Subordination and Hermitization lead to the density in terms of h^{-1}. No parameters are fit. Good.

The soft spots: First, the abstract and Theorem 1.8 contain a typo: the density formula has -2/s - 2β/s^2, which should be -2/t - 2β/t^2. Theorem 6.4 has the correct version, so it's a copy-paste error, but in a theorem statement it matters.

Second, Proposition 5.2, the positivity of the Jacobian, is the load-bearing step, and as printed the proof has a real problem. The stress-test note is right: the row operation r2 := r2 - (β/δ)r3 does not produce the simplified second row from the displayed matrix. But the issue is not the method — it's a sign error in the displayed third row. Differentiating bH3 = δ - pδT/D with respect to α gives -δ T_α D + δ T D_α over D^2, not -δ T_α D - T D_α over D^2. With the plus sign restored, the same row operation works exactly as the text claims, and the subsequent Cauchy-Schwarz argument is plausible. So the gap is a typo, not a fatal flaw — but it needs to be fixed because no one can verify the determinant from the printed version. I'd want to see the corrected calculation or a computer algebra check of det J > 0 before relying on Proposition 5.2.

Third, Assumption 1.6 is conditional for p≤1, and the atom question for p<1 is left open. They're honest about this, and the paper's main result is stated with the assumption. That's fine, but it limits the scope: for p≤1 (except when the sufficient conditions hold) the density formula is conditional.

Fourth, the numerical check is a heuristic; it matches the boundary and density shape, but it doesn't verify the formula. That's okay — the derivation is the argument — but I wouldn't call it confirmation.

The citation pattern looks fine; they cite the relevant subordination and Brown-measure literature, and the self-citations are to prior results in the same line. I don't see a circularity problem: the density is computed from definitions, not fit to the example.

Bottom line: this is a solid, new computation with a clear main theorem, a beautiful worked example, and a proof that's fixable but has typos in a critical place. It deserves peer review. A referee should be asked to verify Proposition 5.2 with the corrected signs and to check whether the positivity can be proved directly. I would not desk-reject. I'd also tell the authors to fix the s/t typo in the abstract and Theorem 1.8 before resubmission.

Referee Report

3 major / 4 minor

Summary. The paper studies the Brown measure of a=i x? of the form x+iy, where x is an arbitrary selfadjoint element and y is a free Poisson element of parameter p, in a tracial W*-probability space. The authors use the matrix-valued Hermitization of a and operator-valued subordination to construct a left inverse H of the subordination function. They define an open set D and a map h:D→M by solving H_{11}(λ,iδ0(λ))=0 and setting h(λ)=H_{12}(λ,iδ0(λ)). The main theorem, Theorem 1.8/6.4, asserts that the Brown measure is supported in cl(M) and that, for s+it∈M, its absolutely continuous density is f=(1/(4π))[(2/t)(∂α/∂s+∂β/∂t)-2/t-2β/t²], where α+iβ=h^{-1}(s+it). The proof proceeds by showing that the logarithmic potential extends real-analytically across M, computing its first derivatives via subordination, and taking the Laplacian. A fully explicit Bernoulli example is worked out, with formulas for D, h, the support curve, and the density, and the support is identified with the spectrum.

Significance. If the main theorem is correct, this is a substantial contribution: it gives a constructive, checkable formula for the absolutely continuous part of the Brown measure for a whole class of non-normal free elements, going beyond the semicircular and elliptic cases. The methodology is not circular: h is constructed explicitly from the left inverse H, no parameters are fitted, and the Bernoulli example is closed-form and agrees with numerical random-matrix simulations. The paper is clearly organized and the subordination framework is natural. The main claims are falsifiable and the explicit example is a valuable sanity check. The main caveat is that one load-bearing Jacobian-positivity proof is only sketched, and the theorem statement contains a typo in the central formula.

major comments (3)

[§5.1, Proposition 5.2] The strict positivity of det J bH at δ0 is the load-bearing point for the inverse-function step in Lemma 6.2 and hence for the real-analyticity of L and the density formula. The proof's final step is only asserted: after the row and column reductions it says 'Expanding along the second column the result follows...'. I checked the row reduction r2 := r2 - (β/δ)r3; it is actually correct, giving row 2 = [-D_α, -D_β, -D_δ]/D². The subsequent column operation c2 := c2 - (β/δ)c3 is also correct. However, the final determinant is not displayed. Writing Q=(α-t)²+β²+δ² and ν=dµ_x(t)/Q², the reduction yields det = 8Tδ²(∫1 dν · ∫(α-t)² dν - (∫(α-t) dν)²)/D⁴, which is nonnegative by Cauchy-Schwarz and positive in the nondegenerate case. This computation should be included explicitly; as written, the proof of a central analytic fact is an unverified calculation.
[§1.3, Theorem 1.8, Eq. (1.9)] The displayed density in the the statement of Theorem 1.8 is f = (1/(4π))[(2/t)(∂α/∂s+∂β/∂t) - 2/s - 2β/s²]. The denominators should be t, not s: the abstract and Theorem 6.4 have -2/t - 2β/t². As stated, readers using Eq. (1.9) will get a wrong density. This must be corrected in the final version.
[§1.3 and §6, Theorems 1.8 and 6.4] Neither theorem lists Assumption 1.6 (spec(x)⊆cl(D)) as a hypothesis, but the proof that the Brown measure has no mass outside cl(M) is Proposition 6.5, which explicitly invokes Assumption 1.6. The abstract acknowledges that conditions on x are needed for some p, but the theorem statements as written appear unconditional. The theorems should either state Assumption 1.6 explicitly or be formulated as conditional statements, with the Section 4 sufficient conditions stated separately.

minor comments (4)

[§6, Lemma 6.1 and Lemma 6.2] In Eq. (6.3) and the displayed formula for L_{x+iy}(z,0), the integrand contains 1/(1+t) when the integration variable is u. It should be 1/(1+u).
[§5.1, Proposition 5.2] The displayed matrices in the proof omit parentheses in expressions like -δ ∂T/∂α D - T ∂D/∂α; this makes the row operations hard to follow. Please write these as -δ(∂T/∂α · D - T · ∂D/∂α) etc.
[Throughout] There are numerous typos: 'operator velued' (§1.5), 'F ree Poisson' (§2.3), 'differmorphism' (§5.3), 'measre' (§1.5), 'exits' (§6, Lemma 6.1). A careful proofreading pass is needed.
[§4, Remark 4.4] The sentence 'We will clarify this in later section' looks like a leftover from an earlier draft and should be removed or replaced by a reference to Proposition 4.3/4.5.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular reduction: Brown-measure density is derived by explicit subordination computation; only a proof gap in Proposition 5.2 and non-load-bearing self-citations.

full rationale

No load-bearing circular step was found. H is defined self-containedly from G_X and the explicit free-Poisson R-transform (Eq. 1.2, Lemmas 2.1-2.3), and D, delta_0, h are defined from H_11 (Definition 1.3), not from the Brown measure. The main formula (Theorem 6.4, Eq. 6.6) is obtained by differentiating L_{x+iy}: Proposition 6.3 computes partial L/partial s = 2(alpha - s)/t and partial L/partial t = 2 beta/t from subordination and Eq. (6.1), and the Laplacian produces Eq. (6.6) directly; no parameter is fitted and no 'prediction' is an input by construction. The support statement is proved, not assumed. The paper cites the authors' own [23], [18], [5], [4], but only as context; the left inverse is explicitly introduced 'in a self-contained way' (Proposition and Definition 1.1). The only serious issue is a proof gap in Proposition 5.2: positivity of the Jacobian is dispatched with the sentence 'Expanding along the second column the result follows after writing explicit formulas for the derivatives from Proposition 5.1 and using Cauchy-Schwarz inequality...' and the displayed row reduction is not fully justified. This threatens completeness of the inverse-function-theorem step, but it is a correctness/verification gap, not a circular reduction: the density formula is not identical to any input by construction. Accordingly the circularity score is 1.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters are fit to data; the only inputs are p and µ_x. The paper relies on standard subordination, Nevanlinna representation, and the Brown-measure calculus. The domain assumption for p≤1 is explicit and flagged.

assumptions (6)

standard math Operator-valued subordination: there exists a Fréchet-analytic Ω with Im Ω(B) ≥ Im B and G_{X+Y}(B) = G_X(Ω(B)) (from [2, Theorem 2.2]).
Invoked in §2.2 to set up the Hermitization; external theorem.
standard math The R-transform of the free Poisson Y is R_Y(B) = p(J−B)^{-1}.
Derived in Lemma 2.1 from free Poisson cumulants; standard fact.
standard math Nevanlinna/Stieltjes integral representation for functions mapping C+ to C−.
Used in Lemma 3.1 and Prop. 1.2 to get the measure ρ.
domain assumption Assumption 1.6: spec(x) ⊆ cl(D).
Imposed for p≤1 (automatic for p>1); needed for extension of h and support containment.
standard math Brown measure is given by µ_a = (2/π) ∂² L_a/∂z∂z̄, L_a = φ(log|z−a|).
Standard definition (Brown/Haagerup-Schultz); used in Theorem 6.4.
standard math Inverse function theorem and Brouwer invariance of domain.
Used in Corollary 5.7 and Section 6.

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Pith. "Pith review of On the Brown measure of $x + i y$, with $x,y$ selfadjoint and $y$ free Poisson." pith.science (2026). https://pith.science/paper/PMAVD2BD

@misc{pith2026251223528,
  author       = {Pith},
  title        = {Pith review of: On the Brown measure of $x + i y$, with $x,y$ selfadjoint and $y$ free Poisson},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PMAVD2BD}},
  note         = {Machine review of arXiv:2512.23528}
}

abstract

Let $x,y$ be freely independent selfadjoint elements in a $W^{*}$-probability space, where $y$ has free Poisson distribution of parameter $p$. We pursue a methodology for computing the absolutely continuous part of the Brown measure of $x + i y$, which relies on the matrix-valued subordination function $\Omega$ of the Hermitization of $x + i y$, and on the fact that $\Omega$ has an explicitly described left inverse $H$. Our main point is that the Brown measure of $x + i y$ becomes more approachable when it is reparametrized via a certain change of variable $h : \mathcal{D} \to \mathcal{M}$, with $\mathcal{D}, \mathcal{M}$ open subsets of $\mathbb{C}$, where $\mathcal{D}$ and $h$ are defined in terms of the aforementioned left inverse $H$, and $\mathrm{cl} \,(\mathcal{M})$ contains the support of the Brown measure. More precisely, we find (with some conditions on the distribution of $x$, which have to be imposed for certain values of the parameter $p$) the following formula: \[ f(s + i \, t) =\frac{1}{4\pi}\left[\frac{2}{t}\left(\frac{\partial \alpha}{\partial s} +\frac{\partial \beta}{\partial t}\right)-\frac{2}{t}-\frac{2\beta}{t^2}\right], \ \ s + i \, t \in \mathcal{M}, \] where $f$ is the density of the absolutely continuous part of the Brown measure and the functions $\alpha, \beta : \mathcal{M} \to \mathbb{R}$ are the real and respectively the imaginary part of $h^{-1}$.

Figures

Figures reproduced from arXiv: 2512.23528 by the authors.

**Figure 2.** The blue line represents the boundary of M. The red dots are the eigenvalues of the matrix approximation XN + iYn. To get the explicit formula for the density one has to invert h(α + iβ) = s + it, that is write α and β as functions of s and t We see that h(α + iβ) = p 4α2 + β 2 + 1 + 4iαβ + β 2α − i . Separating real and imaginary part we arrive at the system of equations    s = 2α p 4α2 + β 2 + 1 − β 4α… view at source ↗

**Figure 3.** Density of the Brown measure of x+iy. Observe that the density has unbounded limits at ±1; this matches how eigenvalues were concentrated around those points in [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

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Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.

[1] [1]

Belinschi and M

Serban T. Belinschi and M. Capitaine,Spectral properties of polynomials in independent Wigner and deterministic matrices, J. Funct. Anal.273(2017), no. 12, 3901–3963. MR 3711884

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Belinschi, Tobias Mai, and Roland Speicher,Analytic subordination theory of operator-valued free additive convolution and the solution of a general random matrix problem, J

Serban T. Belinschi, Tobias Mai, and Roland Speicher,Analytic subordination theory of operator-valued free additive convolution and the solution of a general random matrix problem, J. Reine Angew. Math. 732(2017), 21–53. MR 3717087

2017

[3] [3]

Serban T. Belinschi, Piotr ´Sniady, and Roland Speicher,Eigenvalues of non-Hermitian random matrices and Brown measure of non-normal operators: Hermitian reduction and linearization method, Linear Algebra Appl.537(2018), 48–83. MR 3716236

2018

[4] [4]

Belinschi, Zhi Yin, and Ping Zhong,The Brown measure of a sum of two free random variables, one of which is triangular elliptic, Adv

Serban T. Belinschi, Zhi Yin, and Ping Zhong,The Brown measure of a sum of two free random variables, one of which is triangular elliptic, Adv. Math.441(2024), Paper No. 109562, 62. MR 4710866

2024

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Hari Bercovici and Ping Zhong,The Brown measure of a sum of two free nonselfadjoint random variables, one of which is R-diagonal, 2022, arXiv:2209.12379

arXiv 2022

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Philippe Biane,On the free convolution with a semi-circular distribution, Indiana Univ. Math. J.46 (1997), no. 3, 705–718. MR 1488333

1997

[7] [7]

Math.90(2001), no

Philippe Biane and Franz Lehner,Computation of some examples of Brown ’s spectral measure in free probability, Colloq. Math.90(2001), no. 2, 181–211. MR 1876844

2001

[8] [8]

4, 621–669

Charles Bordenave, Pietro Caputo, and Djalil Chafa ¨ ı,Spectrum of markov generators on sparse random graphs, Communications on Pure and Applied Mathematics67(2014), no. 4, 621–669

2014