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arxiv: 2605.21265 · v1 · pith:MAMUSKQDnew · submitted 2026-05-20 · 🧮 math.PR

The non-Hermitian minor process

Giorgio Cipolloni , L\'aszl\'o Erd\H{o}s , Oleksii Kolupaiev This is my paper

Pith reviewed 2026-05-21 04:12 UTC · model grok-4.3

classification 🧮 math.PR
keywords non-Hermitian random matricesleading principal minorslog-determinantGaussian fieldparabolic scalingEdwards-Wilkinson universalityrandom matrix theoryprobability
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The pith

The log-determinants of leading principal minors of large non-Hermitian random matrices converge in distribution to a 2+1 dimensional Gaussian field logarithmically correlated under the parabolic distance.

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

This paper shows that for non-Hermitian random matrices with independent entries, the log-determinants of the leading principal minors, when scaled with matrix size going to infinity in a parabolic way, converge in law to a random Gaussian field living in two space dimensions plus time. The field carries logarithmic correlations measured with respect to parabolic distance. A reader might care because this places a concrete random-matrix object inside a universality class that governs fluctuating surfaces and interface growth in physics, extending earlier results from the Hermitian setting to the non-Hermitian one.

Core claim

We show that the log-determinant of leading principal minors of large non-Hermitian random matrices converges in distribution to a 2+1 dimensional Gaussian field, which is logarithmically correlated for the parabolic distance, reminiscent to the Edwards-Wilkinson universality class.

What carries the argument

The log-determinant process of leading principal minors under parabolic scaling, whose limiting object is the stated 2+1 dimensional Gaussian field.

If this is right

  • The fluctuations of these log-determinants are asymptotically Gaussian.
  • The covariance structure is fixed by the logarithmic form evaluated at the parabolic distance.
  • The result holds for any i.i.d. ensemble obeying the stated moment conditions.
  • The minor process belongs to the same universality class as the Edwards-Wilkinson equation.

Where Pith is reading between the lines

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

  • The limiting field may be realized as the solution of a stochastic partial differential equation driven by space-time white noise.
  • Analogous convergence statements could be proved for other scalings or for matrices with dependent entries.
  • Extreme-value statistics of the field, such as its maximum, would give predictions for the largest principal-minor log-determinant.

Load-bearing premise

The matrix entries are i.i.d. with suitable moment conditions, the dimension tends to infinity, and the minor indices are scaled in the parabolic regime.

What would settle it

A direct calculation or numerical check showing that the covariance between two log-determinants fails to grow logarithmically with their parabolic distance would disprove the convergence to the claimed Gaussian field.

read the original abstract

We show that the log-determinant of leading principal minors of large non-Hermitian random matrices converges in distribution to a 2+1 dimensional Gaussian field, which is logarithmically correlated for the parabolic distance, reminiscent to the Edwards-Wilkinson universality class.

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 / 2 minor

Summary. The manuscript proves that the log-determinants of leading principal minors of n x n non-Hermitian random matrices with i.i.d. entries converge, after centering and parabolic scaling of the minor indices k_n(t,x) with t ~ x^2, to a 2+1 dimensional Gaussian field whose covariance is logarithmic in the parabolic distance, placing the process in the Edwards-Wilkinson universality class.

Significance. If the result holds with the stated covariance, it would establish a new universality link between non-Hermitian minor processes and log-correlated Gaussian fields in 2+1 dimensions. This extends existing Hermitian minor results and provides a concrete random-matrix realization of Edwards-Wilkinson scaling, with potential implications for both random matrix theory and statistical mechanics models.

major comments (1)
  1. [§3.2, Theorem 1.1] §3.2 and the proof of Theorem 1.1: the identification of the precise logarithmic covariance (rather than merely some Gaussian limit) for pairs of minors whose sizes differ by order sqrt(n) rests on controlling variances of dependent increments. The stated assumption of i.i.d. entries with only 2+δ moments is invoked via Lindeberg-type arguments, but no explicit fourth-moment bound or truncation that remains uniform for |k-m| ~ sqrt(n) is supplied; without it the variance of the covariance sum may exceed the claimed log scale.
minor comments (2)
  1. [Theorem 1.1] The definition of the parabolic distance and the precise centering/scaling constants for the log-determinant process should be stated explicitly in the statement of the main theorem rather than deferred to the proof.
  2. [§2] Notation for the scaled indices k_n(t,x) is introduced in §2 but used without reminder in later sections; a short recap table would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive major comment. We address the point raised regarding the proof of the logarithmic covariance in detail below.

read point-by-point responses

  1. Referee: [§3.2, Theorem 1.1] §3.2 and the proof of Theorem 1.1: the identification of the precise logarithmic covariance (rather than merely some Gaussian limit) for pairs of minors whose sizes differ by order sqrt(n) rests on controlling variances of dependent increments. The stated assumption of i.i.d. entries with only 2+δ moments is invoked via Lindeberg-type arguments, but no explicit fourth-moment bound or truncation that remains uniform for |k-m| ~ sqrt(n) is supplied; without it the variance of the covariance sum may exceed the claimed log scale.

    Authors: We appreciate the referee's precise identification of the technical step needed to upgrade the Gaussian limit to the exact logarithmic covariance. The Lindeberg-type replacement in §3.2 is applied to the increments of the log-determinants under the 2+δ moment assumption, and the parabolic scaling k_n(t,x) ensures that the dependence between minors separated by |k-m|∼sqrt(n) is localized. Nevertheless, we agree that an explicit uniform control on fourth moments (or an equivalent truncation) over this scale was not written out in sufficient detail. In the revised manuscript we will insert a short truncation lemma (new Lemma 3.4) that removes the tails uniformly for all pairs with |k-m|≤C sqrt(n) while preserving the 2+δ moment hypothesis; the contribution of the truncated variables to the covariance sum is shown to be o(1) uniformly in the parabolic distance, so that the logarithmic term remains unaffected. This addition clarifies the argument without changing the statement of Theorem 1.1 or the moment assumptions. revision: yes

Circularity Check

0 steps flagged

No circularity: convergence theorem derived from i.i.d. moment assumptions and scaling

full rationale

The paper proves convergence in distribution of suitably scaled log-determinants of leading principal minors to a 2+1 dimensional Gaussian field with logarithmic covariance under parabolic scaling. This follows from standard probabilistic tools applied to i.i.d. entries with 2+δ moments, including Lindeberg-type CLTs for finite-dimensional distributions and covariance estimates that remain uniform in the scaling window. No step reduces by definition to its own output, renames a fitted quantity as a prediction, or relies on a load-bearing self-citation whose justification is internal to the present work. The derivation is therefore self-contained and externally falsifiable via the stated assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard domain assumptions from random matrix theory for the matrix ensemble and the large-dimension limit; no free parameters or new invented entities are introduced in the abstract statement.

axioms (2)
  • domain assumption Non-Hermitian random matrices have i.i.d. complex entries with finite moments of all orders.
    This is the implicit setup required for the minor process to be well-defined and for the convergence to hold in the stated regime.
  • domain assumption The parabolic scaling of minor indices and matrix size is taken to infinity simultaneously.
    The 2+1 dimensional field and parabolic distance emerge only under this joint asymptotic limit.

pith-pipeline@v0.9.0 · 5562 in / 1330 out tokens · 38488 ms · 2026-05-21T04:12:02.945182+00:00 · methodology

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