Eigenvalues of Brownian Motions on $\mathrm{GL}(N,\mathbb{C})$

F\'elix Parraud; Nicholas A. Cook; Tatiana Brailovskaya; Todd Kemp

REVIEW 2 major objections 5 minor 88 references

Eigenvalues of Brownian Motions on $\mathrm{GL}(N,\mathbb{C})$

T0 review · 2 major / 5 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read For every fixed time, the eigenvalue distribution of Brownian motion on GL(N,C) converges almost surely to the Brown measure of the free multiplicative Brownian motion.

desk verdict Strong paper that likely closes Biane's conjecture at the eigenvalue level, powered by a genuinely new small-time concentration estimate; the key moment bound in Section 3 is condensed and needs referee scrutiny before acceptance. read the letter →

arxiv 2511.10535 v2 pith:24SYJ6ZH submitted 2025-11-13 math.PR math.OA

Tatiana Brailovskaya , Nicholas A. Cook , Todd Kemp , Félix Parraud This is my paper · ORCID

classification math.PRmath.OA MSC 60B2046L5415B52

keywords BrownianmotiononGL(NC)freemultiplicativeBrownmeasureeigenvalueconvergenceHermitizationnon-normalrandommatricessingularvalueconcentrationunitarilyinvariantdiffusions

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 claims that, for every fixed time t, the random eigenvalues of a broad family of Brownian motions on the complex general linear group GL(N,C) — not just traces of polynomial functions, but the full empirical eigenvalue density — converge almost surely to a fixed, explicitly identifiable probability measure. That limit is the Brown measure of the free multiplicative Brownian motion b0b(t), a free-probability object built from freely independent semicircular Brownian motions. The result covers all unitarily invariant ("elliptic") Brownian motions on GL(N,C), parametrized by ϱ>0 and ζ with |ζ|<ϱ, and any uniformly bounded invertible initial condition with its own large-N limit. It closes a gap left open since 1997, where convergence of trace moments was known but convergence of eigenvalues — much harder for non-normal matrices — was not. The proof's engine is a new quantitative statement: for small times, the multiplicative Brownian motion is very close in operator norm to a single additive Gaussian increment, with failure probability at most (Ct/δ)^{N^{2/3}}.

What carries the argument

The small-time affine approximation of Theorem 1.7 — B(t) ≈ I+W(t) with tail (Ct/δ)^{N^{2/3}} — is the load-bearing mechanism. It is proved by first replacing B(t) by a dyadic product of i.i.d. Gaussian increments (a Donsker-type random walk), proving an N-independent L^2 convergence rate for the approximation, and then using an iteration of Gaussian integration by parts (Schwinger–Dyson equations) and a borrowed asymptotic moment expansion to compare the product to a single increment. The N^{2/3} exponent arises from a trace moment bound E tr_N |B(t)−I−W(t)|^{2k} ≤ (tC)^{2k} exp(C k^4/N^2), choosing k ~ N^{2/3}. This one mechanism converts the curved, non-normal diffusion into an additive G

What would settle it

Directly test the small-time tail bound for small N: for N=8 and t=0.1, estimate P(||B(t)−I−W(t)|| ≥ δ) and check whether it is bounded by (Ct/δ)^{N^{2/3}} with a numerical constant C. Alternatively, simulate B_0^N B^N(t) for N=2000 with a fixed invertible initial condition and a large time t, and compare the empirical eigenvalue histogram with the analytic boundary and density of the Brown measure of b0b(t); a persistent mismatch away from the boundary would contradict Theorem 1.6 as stated.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.6: let B_{ϱ,ζ}^N be any nondegenerate, unitarily invariant Brownian motion on GL(N,C), let B_0^N be independent, invertible, uniformly bounded, and converging almost surely in *-distribution to b0, and let b(t) be the free multiplicative Brownian motion. Then almost surely the empirical eigenvalue measure of B_0^N B^N(t) converges weakly to the Brown measure of b0b(t). The proof follows Girko's Hermitization: convergence of traces gives convergence of singular-value measures of shifted matrices, and the new ingredient is enough control of the small singular values to make the log-integral pass from smooth test functions to log. This control comes from Theorem 1

Load-bearing premise

The whole proof leans on an asymptotic expansion of Gaussian-matrix moments quoted without proof from earlier work; if that expansion, applied to |B_n^N(t)−I−W^N(⌊2^n t⌋/2^n)|^{2k}, fails to yield the claimed k^4/N^2 error term, the small-time concentration theorem — and with it the eigenvalue convergence — collapses; the initial condition also requires a uniform spectral gap K^{-1} ≤ σ_min(B_0^N) ≤ σ_max(B_0^N) ≤ K.

Editorial extensions

If this is right

The scattered eigenvalue cloud of B_0^N B^N(t) has, for every fixed time, a deterministic limiting density that can in principle be computed by solving the known PDE formulas for the Brown measure of the free multiplicative Brownian motion.
The same Hermitization machinery yields quantitative singular-value control at mesoscopic scales: the smallest singular values of B(t)−zI are bounded below by polynomial-in-N quantities with high probability, for every z in the plane.
The result validates finite-N eigenvalue simulations as evidence about free multiplicative Brownian motion: for N=2000 the eigenvalue scatter tracks the analytic support boundaries of the Brown measure.
It completes the large-N spectral theory of the Segal–Bargmann–Hall construction: both trace-level and eigenvalue-level convergence now hold for the heat kernel measures on GL(N,C).

Reading between the lines

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

A natural extension is to probe the edge of the limiting eigenvalue cloud at scales N^{-2/3} rather than N^{-1/2}: the N^{2/3} concentration exponent suggests that outlier or boundary-rigidity thresholds for non-normal multiplicative processes may differ from the Hermitian random matrix scale.
The small-time affine approximation is a template that should transfer to other unitarily invariant matrix diffusions and matrix random walks: the essential requirement is an N-independent dyadic L^2 approximation rate, after which Gaussian anti-concentration tools can be imported into multiplicative problems.
If the moment-expansion lemma that supplies the k^4/N^2 control were made fully self-contained, it would likely improve the mesoscopic scale N^{-2/11} in the paper's alternative Wegner estimate and sharpen the constants in the Hermitization argument.
The theorem identifies a deterministic limit but does not compute its density for general non-unitary initial conditions; the paper's stated support conjecture becomes directly testable by simulating the same initial conditions at large N and comparing the cloud boundary with the predicted conformal image region.

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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

This is the real thing. The paper proves almost-sure convergence of the empirical eigenvalue law of Brownian motion on GL(N,C) to the Brown measure of the free multiplicative Brownian motion, for the full two-parameter family of invariant metrics and general invertible initial conditions. That is exactly Biane's conjecture, not the weaker *-distribution convergence that previous work established. The gap between traces of polynomials and eigenvalues of non-normal matrices is real, and the authors are honest about it.

What's new is the small-time approximation: the Euclidean increment I+W(t) captures the multiplicative Brownian motion B(t) up to an error with exponentially small probability, with a precise N^{2/3} exponent. That concentration bound is the engine that powers the Hermitization machinery, and it is a substantial technical contribution on its own. The overall architecture — *-distribution convergence, small-singular-value control, then Bordenave–Chafai style log-potential convergence — is coherent and standard in the good sense. No fitting, no invented entities; the limit is the Brown measure constructed by Brown's method. The simulations are presented as numerical evidence, not proof, which is appropriate.

Soft spots: the proof of Theorem 3.5, the moment bound behind the N^{2/3} exponent, is genuinely compressed. Step 2 (pp. 34–37) asserts rather than demonstrates the key counting inequality 2r1+r2 ≥ 4k−4i and the resulting k^4 exponent; 'one also has similar formulas' covers a lot of differentiations. The stress-test note's worry that a k^5 exponent would break the whole theorem is not paranoid. The argument may well be right, but the text does not give a checkable proof of the constant bookkeeping. A referee who knows Parraud's asymptotic expansion and this kind of counting needs to verify this carefully. The other soft spot is small: the abstract says 'fully resolves' Biane's conjecture, but Biane's quoted phrase was 'limit in distribution,' which is trace-level; the eigenvalue-law reading is standard, but a precise citation would be cleaner.

This is a serious paper for the random matrix/free probability crowd. I'd bring it to reading group, and I'd cite it as soon as it's validated. Send it to referees who know the GUE-moment technology; the recommendation is accept after the Section 3 estimate is pinned down.

Referee Report

2 major / 5 minor

Summary. The paper proves that, for every ρ>0 and complex ζ with |ζ|<ρ, and for any sequence of random initial conditions B_0^N that are a.s. invertible, satisfy the uniform spectral-gap condition (1.13), and converge a.s. in *-distribution to b_0, the empirical eigenvalue law of B_0^N B^N(t) converges weakly almost surely to the Brown measure μ_{b_0 b(t)} of the free multiplicative Brownian motion. The proof combines: (i) a Donsker-type product approximation with uniform L^2 error O(2^{-n/2}) (Proposition 3.3); (ii) a moment estimate (1.14) giving N^{2/3} small-time concentration of B^N(t)-I-W^N(t) (Theorem 1.7); (iii) small-singular-value estimates for shifted products (Proposition 4.1); and (iv) the Bordenave–Chafaï Hermitization lemma (Lemma 6.1). A complementary analytic Wegner estimate is proved in Section 5 for identity initial condition.

Significance. If the technical estimates hold, this is a major advance: it resolves Biane's 1997 conjecture and gives the first eigenvalue-level (not merely trace-level) convergence for a curved non-normal matrix diffusion, with an explicit deterministic limit. The overall architecture is coherent and genuinely novel, particularly the small-time affine approximation theorem, which is of independent interest. The limit measure is not fitted: it is independently defined by free stochastic calculus. The paper also carefully separates the main Hermitization route from the alternative analytic Section 5. My concern is concentrated in one load-bearing place: the proof of the moment estimate (1.14), whose exponent in k is not made checkable.

major comments (2)

[Section 2.3, Lemma 2.25 and its use in (3.4)] The bound (1.14) and hence Theorem 1.7, Proposition 4.1, and the Hermitization in Section 6 all rest on the Step 2 claim that (L^{T_i}...L^{T_1})(Q_n) is a linear combination of at most 2^{4i}(2k)^{4i}(floor(2^n t))^{2i} terms with the stated coefficients, and on the inequality 2r_1+r_2 ≥ 4k−4i. The text asserts both without proof: the first after 'Consequently' and the second as 'then necessarily'. This is not a cosmetic omission. The optimized choice k∼N^{2/3} means the k^4 in exp(C k^4/N^2) is exactly what makes the Markov argument yield N^{2/3} concentration; if the true count were k^{5i}, the optimization would give exp(C N^{4/3}) and the proof would collapse. I am not asserting the estimate is false, but the manuscript does not supply a checkable proof. Please either turn this into a stated combinatorial lemma with proof, or give the full bookkeeping. In addition, the sentence 'oth
[Section 2.3, Lemma 2.25 and its use in (3.4)] Lemma 2.25 is imported from [74] and the sets J_n are explicitly not defined in the paper ('their construction is quite lengthy and will not be used in this paper'). The lemma is stated for a fixed finite d-tuple of variables, but in Theorem 3.5 it is applied to Q_n, a polynomial in O(2^n) variables. What is needed is a version whose constants are uniform as d,n grow, or an explicit reduction to the fixed-d statement. This uniformity is part of the same load-bearing Step 1/Step 2 estimate of (1.14), so it should be addressed as part of the proof, not merely by referring to [74].

minor comments (5)

[Section 5.3] The first paragraph says 'combine Corollary 5.5 with Lemma 5.5'; it should be Lemma 5.6. Also, Section 6 contains a typo 'Lebsegue'.
[References] References [30] and [31] appear to be the same entry (same title, same journal, same volume and year). Please check the intended second paper.
[Abstract and Theorem 1.7] The abstract states a constant C=C_ρ for t∈[0,1], while Theorem 1.7 states C=C(ρ,T) for t∈[0,T]. Please align the two statements.
[Section 3, Step 2] At the beginning of Step 2, the displayed expression for Q_n looks like the polynomial B_n(t)-I-W_N, not |B_n(t)-I-W_N|^{2k}. Clarify the notation, even if the subsequent argument implicitly handles the 2k-th power.
[Proof of Theorem 1.7] The displayed definition of c_{ρ,T} after the Markov step is garbled in the manuscript; please rewrite the sup expression cleanly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the limiting Brown measure is independently defined and the eigenvalue convergence is derived, not assumed.

full rationale

I reviewed the derivation chain from Theorem 1.6 back through Hermitization (Section 6), the singular-value estimates (Proposition 4.1), the small-time concentration estimate (Theorem 1.7), and the moment bound (Theorem 3.5). The candidate limit is the Brown measure of the free multiplicative Brownian motion b0b(t), an object constructed independently via the free SDE (1.17), not defined in terms of the finite-N eigenvalue empirical measures. No parameter is fitted to eigenvalue data, and no eigenvalue-convergence statement is imported as an assumption. The proof of Theorem 1.6 uses the standard Girko Hermitization scheme: Lemma 1.9 verifies *-distribution convergence, relying on external results [24,59,11] and asymptotic freeness; Theorem 3.5 establishes the moment estimate (1.14); Theorem 1.7 converts it into probability concentration; Proposition 4.1 controls small singular values; Section 6 finishes via uniform integrability of the log. The most delicate imported ingredient is Lemma 2.25, quoted from Parraud [74]. This is a substantial citation to a coauthor's earlier published asymptotic expansion, and the present paper does not reproduce its proof. However, that earlier theorem is not the paper's target result, does not assume eigenvalue convergence of GL(N,C) Brownian motions, and is used as a general moment-expansion tool for Gaussian matrices and free semicircular variables. A heavy reliance on prior work, even overlapping prior work, is a completeness/verification concern, not circularity: the steps do not reduce Eq. (1.14), Theorem 1.7, or Theorem 1.6 to their own conclusions. I therefore find no circular step and assign score 0.

Assumptions & free parameters 0 free parameters · 9 assumptions · 0 invented entities

No free parameters: (ρ, ζ) are model parameters of the invariant metric, and all constants (C_ρ, c_ρ, C_{ρ,T}, K) are universal bounds, none fitted to data. The central claim rests on a stack of cited machinery: Parraud's asymptotic expansion ([74]), shifted-Ginibre anti-concentration ([10, 29]), and the Brown-measure regularity results of [37, 50, 51] — the latter partially overlapping with the present authors (Kemp co-authored [59, 37]; Parraud authored [74]). These are published, parameter-free derivations, so the circularity burden is moderate, not extreme. No invented entities are postulated.

assumptions (9)

standard math Standard SDE/stochastic-calculus toolkit for Lie-group-valued Brownian motion (Stratonovich–Itô conversion (1.3)–(1.5), independent multiplicative increments, inverse process (1.5))
Invoked throughout §1.1–1.2 and used in (4.3) for the increment decomposition B(t) d= B(t−ε) B̃(ε).
standard math Free probability framework (W*-probability spaces, free independence, semicircular systems, Voiculescu asymptotic freeness, Proposition 2.7)
Used in §2.1–2.2 and in the proof of Lemma 1.9 to pass from matrix products to the free product b_0 b(t).
domain assumption Asymptotic expansion of Gaussian-matrix moments (Lemma 2.25, quoted from Parraud [74, Lem. 3.6 & Prop. 3.7])
Load-bearing for the central moment estimate Theorem 3.5; not proved in this paper, only cited.
domain assumption Shifted-Ginibre small-singular-value estimates (Lemma 4.3, from Banks–Kulkarni–Mukherjee–Srivastava [10, Lem. 3.3] and Cook [29, Lem. 8.4])
Converts the small-time approximation into quantitative singular-value control in Proposition 4.1.
standard math Hermitization equivalence: the log integral of shifted singular values equals (1/N) log|det(B−zI)|, and the Bordenave–Chafai Hermitization lemma (Lemma 6.1 = [20, Lem. 4.3])
Framework of §1.5 and §6; standard published results that the paper uses without reproof.
standard math Brown measure theory (Brown [23]): subharmonic log-potential whose Laplacian is a probability measure
Defines the candidate limit measure in §1.5 and the target μ_{b_0 b(t)} in the full Theorem 1.6.
domain assumption Strong convergence of the 3-parameter elliptic Brownian motions to the free process (Banna–Capitaine–Cébron [11])
Used in the discussion after Lemma 1.9 and in §5.3 for the uniform-in-N norm bound on B^N(t,ζ).
domain assumption Computed regularity of the Brown measure of the free multiplicative Brownian motion ([37, 53, 50, 51]): support region, smooth density, and the analytic continuation of S(t,z,0,η) through η=0 ([37, Thm 6.4])
Used in §1.6 to describe the limit measure and in Lemma 5.6 for the free Wegner estimate underpinning Proposition 5.1.
domain assumption Hypothesis (1.13): uniform boundedness and spectral gap for the initial condition B^N_0
Explicit hypothesis of Theorem 1.6; used in Proposition 4.1 and §6 through σ_min(B_0) ≥ κ.

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Pith. "Pith review of Eigenvalues of Brownian Motions on $\mathrm{GL}(N,\mathbb{C})$." pith.science (2026). https://pith.science/paper/24SYJ6ZH

@misc{pith2026251110535,
  author       = {Pith},
  title        = {Pith review of: Eigenvalues of Brownian Motions on $\mathrmGL(N,\mathbbC)$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/24SYJ6ZH}},
  note         = {Machine review of arXiv:2511.10535}
}

abstract

We prove that the empirical law of eigenvalues of Brownian motion on the Lie Group $\mathrm{GL}(N,\mathbb{C})$ converges almost surely to a deterministic probability measure, characterized by a free stochastic differential equation. This fully resolves a conjecture made by Philippe Biane in 1997. Our analysis includes a family $\{B=B_{\rho,\zeta}\colon |\zeta|<\rho\}$ of nondegenerate diffusion processes on $\mathrm{GL}(N,\mathbb{C})$ whose laws are invariant under unitary conjugation, with initial distributions assumed to be uniformly bounded and invertible. The crux of our analysis is a strong quantitative approximation of Brownian motion $B(t)$ on $\mathrm{GL}(N,\mathbb{C})$ for small $t$ by a single increment $I+W(t)$, where $W=W_{\rho,\zeta}$ is an elliptic Brownian motion in the Lie algebra $\mathfrak{gl}(N,\mathbb{C}) = \mathbb{M}_N(\mathbb{C})$. Specifically, for any $t\in[0,1]$ and $\delta>0$, \[ \mathbb{P}\left(\|B(t)-I-W(t)\|\geq \delta\right)\leq \left(C t/\delta\right)^{N^{2/3}} \] for a constant $C=C_\rho$. Leveraging independence of multiplicative increments of the Brownian motion then allows us to use powerful (anti-)concentration tools for Gaussian matrices to complete the Hermitization procedure for convergence of eigenvalues.

Figures

Figures reproduced from arXiv: 2511.10535 by the authors.

**Figure 1.** Eigenvalues of BN 0 BN (1) with N = 2000 and parameters (ϱ, ζ) = (2, 0.6 + i), with two different initial conditions. Also plotted is the boundary of Supp µb0,t [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗

**Figure 2.** Eigenvalues of BN 1,0 (t) with N = 2000 and t = 3 (left) and t = 4 (right). Also plotted is the boundary ∂Σ(1, t), cf. (1.27). ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● … view at source ↗

**Figure 3.** Eigenvalues of BN 0 BN 1,0 (t) with N = 2000 and t = 2 3 (left) and t = 0.7 (right), with initial condition BN 0 a unitary u6 with equal mass eigenvalues at the 6th roots of unity (highlighted in the figures). Also plotted is the boundary ∂Σ(u6, t), cf. (1.27). Following this, in [50] Hall–Ho refined the PDE analysis from the above-mentioned papers to allow the Brownian motions bϱ,ζ (t) from the full parameter range… view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: Eigenvalues of BN 0 BN (t, ζ) with N = 2000. On the left, BN 0 = I and (t, ζ) = (3, 2 − i); on the right, BN 0 is unitary with 6th roots of unity as eigenvalues, and (t, ζ) = (2/3, −i/3). Also plotted are the boundaries ∂Σ(b0, t, ζ), cf. Theorem 1.13. refining those id…

**Figure 5.** Figure 5: Eigenvalues of BN 0 BN (t, ζ) with N = 2000. From top to bottom, t ranges through 0.8, 1, 1.2; ζ = 0 on the left and ζ = 0.5 on the right. In all cases, BN 0 is a normal matrix with empirical eigenvalue distribution 1 8 (δ1 + δ−1 + δ3 + δ−3) + 1 4 (δi + δ−i). Also plot…

**Figure 6.** Figure 6: Eigenvalues of BN 0 BN (1, ζ) with N = 2000. The initial condition is the non-normal 2 × 2 matrix B N 0 = b0 = 1 + i 1 0 −1 + i . On the left, ζ = 0; on the right, ζ = 0.5 + 0.5i. Also plotted are the boundaries ∂Σ(b0, 1, ζ) from (1.31) [PITH_FULL_IMAGE:figures/fu…

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