arxiv: 2604.23708 · v1 · submitted 2026-04-26 · 🧮 math-ph · cond-mat.dis-nn· math.MP· math.PR

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Gradual eigenvector ergodization in coupled Ginibre matrices

Margherita Disertori , Yan V. Fyodorov

Authors on Pith no claims yet

Pith reviewed 2026-05-08 05:08 UTC · model grok-4.3

classification 🧮 math-ph cond-mat.dis-nnmath.MPmath.PR

keywords coupled Ginibre matriceseigenvector ergodizationnon-Hermitian random matriceseigenvalue densitylarge-N asymptoticstransition regimedissipative quantum chaos

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The pith

Eigenvectors of two coupled complex Ginibre matrices spread gradually over the full system as coupling strength |c| grows, according to an explicit asymptotic formula in the large-N limit.

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

The paper examines a model consisting of two independent N by N complex Ginibre matrices coupled by the deterministic term c times the identity matrix. Eigenvectors begin localized within one of the two matrices at zero coupling and progressively delocalize across both matrices as the magnitude of the coupling parameter |c| increases. The authors obtain an explicit asymptotic formula that describes the eigenvector spread in the limit of large matrix size N, covering the entire transition from localized to ergodic behavior. A byproduct calculation yields the mean eigenvalue density at the origin of the spectral bulk in the regime of strong coupling scaled as c equals square root of N times tilde c; this density vanishes beyond the critical value of absolute value tilde c equals 1, indicating a split of the support into two disjoint domains in the complex plane.

Core claim

The paper establishes that the eigenvectors of the coupled Ginibre ensemble undergo gradual ergodization controlled by the coupling parameter |c|, and that this process admits a very explicit asymptotic description in the N to infinity limit that supplies a complete quantitative picture of the spread throughout the transition regime. As a by-product of the same analysis, the mean eigenvalue density at z equals 0 is computed in the fully ergodic regime with scaled coupling c equals square root of N times tilde c and is shown to vanish for absolute value tilde c greater than 1 as N tends to infinity, signalling the splitting of the density support into two disjoint domains.

What carries the argument

The two-matrix ensemble formed by independent complex Ginibre matrices coupled through the deterministic term c times the identity, whose eigenvector localization properties are extracted via large-N asymptotic analysis.

If this is right

The eigenvector spread can be tracked with an explicit formula for any value of the coupling |c| throughout the transition regime.
In the scaled strongly coupled regime the eigenvalue density at the origin drops to zero beyond absolute value tilde c equals 1, causing the support to split into two separate domains in the complex plane.
The results supply a quantitative description of eigenvector delocalization applicable to non-Hermitian models of dissipative quantum systems.

Where Pith is reading between the lines

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

The same asymptotic techniques could be applied to other block-structured non-Hermitian ensembles to detect analogous gradual transitions.
The critical coupling value at which the density splits may mark a change in the stability or localization properties of the spectrum that could be probed in numerical experiments at moderate matrix sizes.
Physical realizations with tunable gain and loss, such as open optical systems, might exhibit measurable signatures of the predicted gradual eigenvector spreading.

Load-bearing premise

The quantitative formulas hold in the strict limit of infinite matrix size N for the specific model of two independent Ginibre matrices with coupling exactly proportional to the identity.

What would settle it

Numerical diagonalization of finite but large coupled matrices that measures the average eigenvector overlap with each individual subsystem as a function of |c| and checks whether the values approach the predicted asymptotic expression, or that computes the eigenvalue density at the origin for scaled couplings and verifies whether it approaches zero above the critical value absolute value tilde c equals 1.

read the original abstract

Non-Hermitian random matrices provide a useful framework for understanding universal characteristics of dissipative quantum chaotic systems with loss or gain. We consider a model of two such system represented by two independent $N\times N$ complex Ginibre matrices interacting via a deterministic matrix $c{\bf 1}_N$, where $c$ is the complex coupling parameter whose magnitude $|c|$ controls the interaction strength. We characterize quantitatively how the eigenvectors of the whole system, initially localized in one of the individual subsystems for $|c|=0$, eventually spread over the full system with growing interaction strength. The resulting asymptotic formula describing such spread in the limit $N\to \infty$ is very explicit and provides a full picture of the gradual ergodization of eigenvectors as a function of the coupling parameter $|c|$ in the whole transition regime. As a by-product of our method we also compute the mean eigenvalue density for our model at the origin of the spectral bulk $z=0$ in the fully ergodic regime, when the coupling is scaled with the matrix size as $c=\sqrt{N}\tilde{c}$. We find that as $N\to \infty$ the limiting density at the origin vanishes beyond the critical value $|\tilde{c}|=1$, signalling of a split of the density support in the complex plane into two disjoint domains.

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

Explicit large-N formula for eigenvector spread in coupled Ginibres plus density split at critical coupling.

read the letter

The main takeaway is that this work gives a very explicit large-N asymptotic formula for the gradual ergodization of eigenvectors in two Ginibre matrices coupled by a deterministic all-ones term, plus a density result showing support splitting at critical coupling. They model the system as two independent complex Ginibre matrices plus c 1_N, then track how the eigenvectors, localized at zero coupling, spread over the whole system as |c| grows. The formula covers the entire transition regime without gaps. As a side calculation they find the mean density at z=0 goes to zero for scaled coupling above 1, which splits the support. This is solid because it provides quantitative control in a clean non-Hermitian setup that connects to dissipative chaos models. The approach seems to rely on standard large-N techniques for these matrices, and the results come out directly without hidden parameters or circular steps. The soft spots are minor: everything is strictly in the N to infinity limit, so one would want to see how well it approximates finite matrices in practice. The specific form of the coupling makes the calculation feasible but means it is not immediately general. The density vanishing is presented as a byproduct, which is fine but not the core. This paper is for people working on random matrix models of open quantum systems or eigenvector statistics in non-Hermitian ensembles. A reader looking for concrete formulas to compare with numerics or experiments in dissipative systems would get real value. It deserves a serious referee because the claims are new, internally consistent, and grounded in the model. I recommend sending it for peer review.

Referee Report

0 major / 3 minor

Summary. The paper considers two independent N×N complex Ginibre matrices coupled by the deterministic all-ones matrix c 1_N. It derives an explicit large-N asymptotic formula for the gradual spread of eigenvectors from one subsystem to the full system as the coupling strength |c| increases through the transition regime. As a by-product, in the fully ergodic scaling c = √N tilde{c}, the mean eigenvalue density at the bulk origin z=0 is computed and shown to vanish for |tilde{c}| > 1, indicating a splitting of the support into two disjoint domains.

Significance. If the derivations hold, the explicit asymptotic formula supplies a complete quantitative description of eigenvector ergodization across the entire transition in |c|, which is a useful addition to the literature on non-Hermitian random matrices and dissipative quantum chaos. The identification of the critical value |tilde{c}|=1 for the vanishing of the density at z=0 is a concrete, falsifiable prediction that follows directly from the large-N analysis of the coupled model.

minor comments (3)

[Abstract] Abstract: the phrase 'the whole transition regime' is used without an explicit interval for |c|; a brief statement of the scaling regimes (e.g., |c| = O(1) versus the √N scaling) would improve clarity for readers.
[Introduction] The manuscript should include a short paragraph in the introduction or methods section stating the precise definition of eigenvector spread (e.g., via the squared overlap with each subsystem or the participation ratio) that is being tracked in the asymptotic formula.
[Density computation section] In the eigenvalue-density calculation, the error control or rate of convergence to the limiting density at z=0 should be stated explicitly, even if only heuristically, to support the claim that the density vanishes beyond |tilde{c}|=1.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive and accurate summary of our manuscript on gradual eigenvector ergodization in coupled Ginibre matrices, including the explicit large-N asymptotic formula and the vanishing of the mean eigenvalue density at z=0 for |tilde c| > 1. We appreciate the assessment of significance and the recommendation for minor revision. No specific major comments were listed in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper defines a coupled-Ginibre model with deterministic all-ones interaction and performs a strict large-N asymptotic analysis to obtain explicit formulas for eigenvector spread across the full |c| transition and the bulk density at z=0 (vanishing for |tilde c|>1). All quantitative results are stated as direct consequences of the model and the N→∞ limit; no equations reduce by construction to fitted inputs, self-referential definitions, or load-bearing self-citations. The abstract and claimed results contain no ansatz smuggling, uniqueness theorems imported from the authors' prior work, or renaming of known patterns as new derivations. The chain is therefore independent of the target outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the independence of the two Ginibre matrices, the deterministic coupling form, and the validity of the large-N asymptotic regime; no additional free parameters or invented entities are introduced in the abstract.

axioms (2)

domain assumption The two N×N matrices are independent complex Ginibre ensembles.
Explicitly stated as the starting model in the abstract.
standard math Asymptotic formulas hold in the strict limit N→∞.
Standard assumption in random-matrix theory for obtaining explicit limiting expressions.

pith-pipeline@v0.9.0 · 5540 in / 1381 out tokens · 74788 ms · 2026-05-08T05:08:59.268421+00:00 · methodology

discussion (0)

Reference graph

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