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arxiv: 2604.13134 · v1 · submitted 2026-04-14 · 🧮 math.PR

From Gaussian to Gumbel: extreme eigenvalues of complex Ginibre products with exact rates

Yutao ma , Xujia Meng This is my paper

Pith reviewed 2026-05-10 15:47 UTC · model grok-4.3

classification 🧮 math.PR
keywords complex Ginibre matricesproduct of random matricesextreme eigenvaluesspectral radiusGumbel distributionGaussian limitdeterminantal point processesconvergence rates
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The pith

The spectral radius of products of complex Ginibre matrices converges after rescaling to a one-parameter family of distributions that continuously interpolates between the standard normal and Gumbel laws.

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

The paper studies the product of k_n independent n-by-n complex Ginibre matrices and tracks the behavior of its largest-modulus eigenvalue as n grows. With alpha defined as the limit of n over k_n, the rescaled spectral radius converges in distribution to a nontrivial law Phi_alpha when alpha lies in (0, infinity), to the Gumbel distribution when alpha is infinity, and to the standard normal when alpha is zero. The family Phi_alpha extends continuously to the two boundary regimes, so the three limiting cases are joined by a single parameter. Exact rates of convergence are obtained in every regime. The same continuous interpolation appears for the rightmost real part, though the limiting law itself is not given in closed form when alpha is finite and positive.

Core claim

After suitable centering and scaling, the spectral radius max |Z_j| converges weakly to Phi_alpha for alpha in (0, infinity), to the Gumbel distribution when alpha equals infinity, and to the standard normal when alpha equals zero. The family of laws Phi_alpha extends continuously at the boundaries: Phi_alpha tends to the normal law as alpha tends to zero from above and to the Gumbel law as alpha tends to infinity. Exact rates of convergence are derived for the spectral radius in the fixed-alpha regime and at both boundaries. For the rightmost real part, convergence rates are established at the boundaries while the interior case is shown to interpolate continuously between normal and Gumbel.

What carries the argument

Determinantal point process representation of the eigenvalues, reduced via rotational invariance to a product of Gamma tail probabilities for the modulus and via polar-coordinate decomposition involving an independent uniform angle plus trigonometric integrals for the real part.

If this is right

  • The three classical extreme-value regimes for Ginibre products are no longer separate cases but form the continuous boundary and interior of a single one-parameter family.
  • Exact rates of convergence are available both when alpha is held fixed and when it tends to the boundaries.
  • The real-part extreme follows the same continuous interpolation between normal and Gumbel limits, even though its interior limiting law lacks a closed-form expression.
  • The transition occurs uniformly in the product length k_n as long as the ratio n/k_n converges.

Where Pith is reading between the lines

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

  • Varying the number of factors in the product can be used to tune the tail of the largest eigenvalue from light (normal) to heavy (Gumbel).
  • The same reduction technique may apply to other products of random matrices whose eigenvalues admit a determinantal description.
  • The continuous family Phi_alpha supplies a natural interpolation that could be tested against finite-n data for intermediate values of alpha.

Load-bearing premise

The determinantal point process representation of the eigenvalues remains valid and permits reduction to n-by-n determinants uniformly across all regimes of alpha.

What would settle it

A numerical computation for large n with n/k_n approaching a fixed positive alpha that shows the empirical distribution of the rescaled max |Z_j| failing to approach the predicted Phi_alpha or failing to match the stated convergence rate.

read the original abstract

We consider the product of \(k_{n}\) independent \(n\times n\) complex Ginibre matrices and denote its eigenvalues by \(Z_{1},\ldots ,Z_{n}\). Let \(\alpha = \lim_{n\to\infty} n / k_{n}\). Using the determinantal point process method, we reduce the study of extremal eigenvalues to the evaluation of determinants of certain \(n\times n\) matrices. In the modulus case, rotational invariance makes the relevant matrix diagonal, which yields a product representation in terms of Gamma tail probabilities. In the real-part case, the matrix is no longer diagonal; we handle this by a polar-coordinate reduction that introduces an independent uniform angle and leads to explicit formulas involving Gamma variables and trigonometric integrals. After appropriate rescaling, the spectral radius \(\max_{1\leq j\leq n}|Z_{j}|\) converges weakly to a nontrivial distribution \(\Phi_{\alpha}\) when \(\alpha \in (0, +\infty)\), to the Gumbel distribution when \(\alpha = +\infty\), and to the standard normal distribution when \(\alpha = 0\). The family \(\{\Phi_{\alpha}\}_{\alpha >0}\) extends continuously to the boundary regimes: \(\Phi_{\alpha}\) converges weakly to the standard normal law as \(\alpha \to 0^{+}\) and to the Gumbel law as \(\alpha \to +\infty\). Thus the three limiting regimes are connected by the single parameter \(\alpha\), yielding a continuous transition from Gaussian to Gumbel distribution. For the spectral radius, we obtain the exact rates of convergence both in the fixed-\(\alpha\) regime and at the boundaries \(\alpha = 0\) and \(\alpha = +\infty\). For the rightmost eigenvalue \(\max_{1\leq j\leq n}\Re Z_{j}\), we establish the convergence rates in the boundary regimes, while for \(\alpha \in (0, +\infty)\) we show that the limiting distribution, though not available in closed form, still interpolates continuously between the normal and Gumbel laws.

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

1 major / 1 minor

Summary. The manuscript examines the extreme eigenvalues of the product of k_n n×n complex Ginibre matrices, with α = lim n→∞ n/k_n. Using determinantal point processes, it reduces the problem to n×n matrix determinants. For the spectral radius, this yields a product of Gamma tail probabilities due to rotational invariance. For the real part, a polar reduction leads to Gamma variables and trigonometric integrals. The paper claims that after rescaling, max |Z_j| converges weakly to Φ_α for α ∈ (0,∞), Gumbel for α=∞, and standard normal for α=0, with continuous extension at boundaries, and provides exact convergence rates. Similar results for the rightmost real part, with limiting distribution for fixed α not closed form but interpolating.

Significance. If the derivations hold, this provides a parameterized family of limiting distributions that continuously connects the Gaussian regime (α=0) to the Gumbel regime (α=∞) for extremes in Ginibre products. The explicit reductions to Gamma tails and integrals, along with exact rates, represent a technical advance in extreme value theory for non-Hermitian random matrices. The continuous interpolation is a notable feature that unifies previously separate regimes.

major comments (1)
  1. [Abstract (DPP reduction paragraph)] The reduction of extremal eigenvalue probabilities to determinants of n×n matrices derived from the DPP kernel of the Ginibre product (as described in the abstract) is asserted to hold uniformly when α = lim n/k_n ∈ [0,∞]. However, as k_n grows linearly with n, the kernel involves α-dependent factors (likely Meijer-G or hypergeometric functions). The manuscript must provide explicit uniformity estimates or error bounds to justify interchanging the n→∞ limit with the determinant evaluation for the maximum probability; without this, the claimed continuous family {Φ_α} and its boundary behaviors rest on an unverified limit interchange.
minor comments (1)
  1. [Abstract] The abstract mentions 'exact rates of convergence' for the spectral radius in fixed-α and boundaries; it would be helpful to specify the form of these rates (e.g., in terms of n or log n) for clarity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the constructive comment on the uniformity of the DPP reduction. We clarify below that the reduction is exact for each n, with limits taken subsequently via direct asymptotics, and we will add explicit uniformity estimates in the revision to address the concern fully.

read point-by-point responses

  1. Referee: [Abstract (DPP reduction paragraph)] The reduction of extremal eigenvalue probabilities to determinants of n×n matrices derived from the DPP kernel of the Ginibre product (as described in the abstract) is asserted to hold uniformly when α = lim n/k_n ∈ [0,∞]. However, as k_n grows linearly with n, the kernel involves α-dependent factors (likely Meijer-G or hypergeometric functions). The manuscript must provide explicit uniformity estimates or error bounds to justify interchanging the n→∞ limit with the determinant evaluation for the maximum probability; without this, the claimed continuous family {Φ_α} and its boundary behaviors rest on an unverified limit interchange.

    Authors: We thank the referee for highlighting this technical point. The reduction of the extremal probabilities to n×n determinants is exact for every finite n and every k_n; it follows directly from the determinantal structure of the eigenvalue point process of the Ginibre product, whose correlation kernel is given explicitly in terms of Meijer G-functions (or equivalent hypergeometric representations) with parameters depending on n and k_n. This exact representation holds uniformly in the sense that it requires no limit and is valid for any sequence k_n, including when n/k_n → α ∈ [0,∞]. The subsequent n→∞ analysis is performed on this exact determinant expression. For fixed α ∈ (0,∞), the modulus case reduces to a product of Gamma tail probabilities whose parameters scale with n/k_n; the asymptotics of these tails are derived using Stirling-type expansions that remain valid uniformly for α in any compact interval [δ,M] ⊂ (0,∞). The real-part case is handled analogously after the polar reduction. The limiting distributions Φ_α are obtained from these asymptotics, and the continuous extension to the boundaries is shown by letting α→0 or α→∞ inside the limiting expressions. To make the uniformity fully explicit and to justify any implicit interchange of limits, we will add in the revised manuscript a new lemma providing error bounds on the kernel approximation and on the determinant that are uniform for α ∈ [δ,M]. These bounds will be stated with explicit rates in n and will cover both the fixed-α regime and the passage to the boundary regimes. We believe this addition will resolve the concern while leaving the core derivations unchanged. revision: yes

Circularity Check

0 steps flagged

No circularity: derivations rest on classical DPP kernel for Ginibre products and explicit Gamma tail reductions

full rationale

The paper begins from the established determinantal point process representation of eigenvalues of Ginibre matrix products (a standard fact in random matrix theory, not derived within the paper) and reduces the extremal statistics to n-by-n determinants. For the modulus, rotational invariance diagonalizes the matrix to a product of independent Gamma tail probabilities; for the real part, a polar-coordinate change of variables yields explicit integrals over Gamma variables. These steps are algebraic identities and known distributional facts, not self-referential fits or self-citations that presuppose the target limit laws. The subsequent weak-convergence claims and continuous interpolation in alpha follow from asymptotic analysis of these explicit expressions, with no parameter fitting or renaming of inputs as predictions. The uniformity of the DPP reduction for k_n growing linearly with n is asserted but does not create a definitional loop; it is an external analytic estimate whose validity is independent of the final Phi_alpha family.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claims rest on the determinantal point process property of products of complex Ginibre matrices and on standard tail properties of Gamma random variables; no free parameters are fitted and no new entities are postulated.

axioms (2)
  • domain assumption Eigenvalues of the product of independent complex Ginibre matrices form a determinantal point process.
    Invoked to reduce extremal probabilities to determinants of n-by-n matrices.
  • domain assumption Rotational invariance of the complex Ginibre ensemble makes the modulus matrix diagonal.
    Used to obtain the product representation in Gamma tails.

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