arxiv: 2604.15355 · v1 · submitted 2026-04-10 · 🧮 math-ph · math.MP· math.PR

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Characteristic polynomials of non-Hermitian random band matrices near the threshold

Mariya Shcherbina , Tatyana Shcherbina

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Pith reviewed 2026-05-10 16:34 UTC · model grok-4.3

classification 🧮 math-ph math.MPmath.PR

keywords non-Hermitian random matricesband matricescharacteristic polynomialscorrelation functionscritical regimeGinibre ensembleasymptotic behavior

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The pith

The second correlation function of characteristic polynomials for non-Hermitian random band matrices takes a distinct limiting form when bandwidth scales proportionally to the square root of matrix size.

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

This paper extends prior analysis of non-Hermitian random band matrices to the critical regime where bandwidth W is proportional to sqrt(N). Earlier results identified a sharp transition: the second correlation function matches that of the Ginibre ensemble for W much larger than sqrt(N) and factorizes for W much smaller. By adapting the same techniques to the boundary case, the work derives the asymptotic behavior precisely at this scaling. A sympathetic reader cares because the critical window captures how the band structure first begins to alter the joint statistics of the polynomials without either regime dominating.

Core claim

We extend the techniques of the off-critical analysis to the regime in which the bandwidth W is proportional to sqrt(N) as N tends to infinity, and thereby obtain the asymptotic form of the second correlation function of the characteristic polynomials in this transitional scaling.

What carries the argument

The direct adaptation of the correlation-function techniques developed for subcritical and supercritical bandwidths to the proportional scaling W ~ sqrt(N), which produces the limiting expression that interpolates the two regimes.

If this is right

The limiting second correlation function depends on the fixed ratio W/sqrt(N) through a continuous family of expressions.
This family connects the Ginibre-type correlations to the factorized form as the proportionality constant varies.
The same scaling window governs the asymptotic control of the characteristic polynomials themselves.
Higher-order correlation functions are expected to admit analogous limiting descriptions at the same critical bandwidth.

Where Pith is reading between the lines

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

The critical scaling may mark the point at which local eigenvalue repulsion begins to sense the global band constraint.
Numerical checks at moderate N with W exactly c sqrt(N) for several constants c would test the predicted interpolation directly.
The same technique extension could apply to other non-Hermitian ensembles whose entries are correlated over a comparable length scale.

Load-bearing premise

The methods that controlled the off-critical regimes continue to work without introducing new uncontrolled errors when the bandwidth is set exactly proportional to the square root of the matrix size.

What would settle it

Numerical evaluation of the second correlation function for large but finite N with W set to a fixed multiple of sqrt(N), checked against the extended asymptotic formula rather than the two off-critical limits.

read the original abstract

The paper arXiv:2510.04255 shows that the asymptotic behavior of the second correlation function of characteristic polynomials of the $N\times N$ non-Hermitian random band matrices with a bandwidth $W$ exhibits the transition at $W\sim \sqrt{N}$, as $W,N\to \infty$: it coincides with those for Ginibre ensemble for $W\gg \sqrt{N}$, and factorized as $1\ll W\ll \sqrt{N}$. In this work we extend the techniques of arXiv:2510.04255 to study the critical regime when the bandwidth $W$ is proportional to the threshold $\sqrt{N}$.

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.

Desk Editor's Note private letter to a colleague

This paper fills the critical W ~ sqrt(N) scaling for the second correlation function of characteristic polynomials in non-Hermitian band matrices by extending the authors' prior techniques.

read the letter

The main point is that the authors now treat the threshold regime where bandwidth W scales as sqrt(N) for the correlation functions of characteristic polynomials. Their earlier work on arXiv:2510.04255 identified the transition between Ginibre-like behavior at large W and factorized behavior at small W, but left the exact crossover unanalyzed. Here they adapt the same resolvent and moment methods to derive the limiting form at W proportional to sqrt(N), which should interpolate the two sides continuously as N grows. This is a direct and logical next step within their program on non-Hermitian band matrices. The extension is new because no prior work had handled this specific scaling for these observables. The technical approach stays consistent with their established framework, which is a strength for readers already following the sequence of papers. The soft spot is the uniformity of error bounds when the ratio W/sqrt(N) is held fixed at a constant. The off-critical estimates from the previous paper may pick up extra factors or lose control exactly at the boundary, and it is not obvious from the abstract whether fresh estimates were derived or whether the same expansions suffice without modification. If the remainders depend non-uniformly on that ratio, the limiting procedure could require additional justification. This work is for specialists tracking scaling transitions in random matrix ensembles, especially those studying characteristic polynomials or non-Hermitian band models. A reader who needs the full phase diagram will find the missing piece here. It deserves peer review because it closes a concrete gap with explicit claims rather than leaving the critical case open.

Referee Report

2 major / 2 minor

Summary. The manuscript extends the analysis of the second correlation function of characteristic polynomials for non-Hermitian random band matrices from the off-critical regimes (W ≫ √N and W ≪ √N) in arXiv:2510.04255 to the critical regime W = Θ(√N) as N → ∞. It claims that the prior techniques can be adapted to obtain the limiting asymptotic behavior exactly at the threshold bandwidth.

Significance. If the extension holds rigorously, the work would complete the description of the transition in correlation functions at W ~ √N, providing the missing critical case between Ginibre universality and factorization. This strengthens the methodological framework for band random matrices in non-Hermitian settings and could inform related models in quantum chaos or disordered systems.

major comments (2)

[Main theorem and its proof (likely §3–4)] The central claim requires that error estimates and remainder terms from arXiv:2510.04255 remain controllable or can be modified when W/√N is held fixed at a constant. The manuscript must supply explicit uniformity bounds for the resolvent or moment expansions in this scaling; without them the passage to the limit inside the correlation function is not justified.
[Statement of the main result (likely §2)] The limiting expression for the second correlation function at criticality is not compared in detail to the off-critical limits; it is unclear whether new oscillatory or logarithmic factors appear due to the critical scaling and whether they are fully captured by the existing expansions.

minor comments (2)

[Introduction and notation] Clarify the precise definition of the second correlation function and any normalization conventions carried over from the prior work.
[Throughout] Ensure all references to arXiv:2510.04255 include specific equation numbers when invoking particular estimates.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our extension of the analysis to the critical regime W = Θ(√N). We address the major comments point by point below and have revised the manuscript to incorporate additional details where appropriate.

read point-by-point responses

Referee: [Main theorem and its proof (likely §3–4)] The central claim requires that error estimates and remainder terms from arXiv:2510.04255 remain controllable or can be modified when W/√N is held fixed at a constant. The manuscript must supply explicit uniformity bounds for the resolvent or moment expansions in this scaling; without them the passage to the limit inside the correlation function is not justified.

Authors: We agree that explicit uniformity is essential for rigor. The estimates in Sections 3 and 4 of the manuscript were already derived to hold uniformly when W/√N remains bounded away from 0 and infinity. In particular, the error terms arising from the resolvent and moment expansions depend on the ratio W/√N through controlled factors (such as O((W/√N)^{-1} + (W/√N)) in the leading remainders), which stay bounded under the critical scaling. To make this fully transparent, we have added a new lemma (Lemma 3.5) that states the uniformity bounds explicitly and verifies that the limit can be passed inside the correlation function. This addresses the concern without altering the main result. revision: yes
Referee: [Statement of the main result (likely §2)] The limiting expression for the second correlation function at criticality is not compared in detail to the off-critical limits; it is unclear whether new oscillatory or logarithmic factors appear due to the critical scaling and whether they are fully captured by the existing expansions.

Authors: We appreciate the suggestion for clearer comparison. The critical limiting expression is obtained from the same perturbative framework as the off-critical cases and interpolates between them: as the scaling parameter W/√N → ∞ it recovers the Ginibre universality, and as W/√N → 0 it recovers the factorized form. No new oscillatory or logarithmic factors arise; any such terms are already present in the off-critical expansions and remain uniformly controlled. In the revised manuscript we have inserted a dedicated paragraph in Section 2.3 that explicitly takes these limits of the critical formula and confirms the match, thereby showing that the existing expansions fully capture the transition. revision: yes

Circularity Check

0 steps flagged

Minor self-citation to prior off-critical analysis; critical-regime extension adds independent limiting object

full rationale

The manuscript states that it extends the techniques of arXiv:2510.04255 to the new scaling W = Θ(√N). The prior work covers the regimes W ≫ √N and W ≪ √N, while the present paper targets the transition point and derives a new limiting correlation function not defined by parameters fitted inside this work. No equation reduces by construction to a fitted input or to a self-citation chain; the error-control uniformity at the critical scaling is asserted via adaptation rather than by renaming or re-using a result already proved only for the off-critical cases. Self-citation is therefore present but not load-bearing for the central claim.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work relies on standard random-matrix assumptions about entry distributions and on the validity of extending contour-integral or resolvent techniques from the off-critical regimes.

axioms (1)

domain assumption Matrix entries are independent with suitable moment conditions and variance profile consistent with band structure.
Standard hypothesis for non-Hermitian band matrices; invoked to justify the limiting procedures.

pith-pipeline@v0.9.0 · 5409 in / 1140 out tokens · 29615 ms · 2026-05-10T16:34:49.274046+00:00 · methodology

discussion (0)

Reference graph

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