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arxiv: 2306.00300 · v4 · submitted 2023-06-01 · 🧮 math.PR · cond-mat.stat-mech· math-ph· math.MP

Eigenvalues, eigenvector-overlaps, and regularized Fuglede-Kadison determinant of the non-Hermitian matrix-valued Brownian motion

Syota Esaki , Makoto Katori , Satoshi Yabuoku This is my paper

Pith reviewed 2026-05-24 08:40 UTC · model grok-4.3

classification 🧮 math.PR cond-mat.stat-mechmath-phmath.MP
keywords non-Hermitian matrix Brownian motioneigenvalue processeseigenvector overlapsFuglede-Kadison determinantstochastic differential equationsscale invariancerandom matrix theorypoint processes
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The pith

Non-Hermitian matrix Brownian motion admits scale-invariant SDEs coupling eigenvalues to eigenvector overlaps via a regularized Fuglede-Kadison determinant.

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

The paper derives a system of stochastic differential equations for the joint evolution of the eigenvalue process and the eigenvector-overlap process in a random matrix whose entries are independent complex Brownian motions. Bi-orthogonality between left and right eigenvectors is imposed so that eigenvector rescaling leaves the eigenvalues unchanged while the overlap matrix transforms accordingly. This coupled system is shown to be invariant under such scale transformations. The regularized Fuglede-Kadison determinant, defined with an auxiliary complex variable, yields stochastic partial differential equations whose logarithmic derivatives recover the time-dependent eigenvalue point process weighted by the diagonal elements of the overlap matrix. Averaging those SPDEs produces deterministic PDEs for the expected densities.

Core claim

The coupled eigenvalue and eigenvector-overlap processes satisfy SDEs whose form is preserved under scale transformations of the eigenvectors; the regularized Fuglede-Kadison determinant supplies an auxiliary random field whose derivatives encode the weighted eigenvalue point process, and averaging the associated SPDEs recovers deterministic evolution equations for the expectations.

What carries the argument

The eigenvector-overlap process (the Hermitian matrix whose entries are products of right-eigenvector and left-eigenvector overlaps) coupled to the eigenvalue process through the derived SDE system, together with the auxiliary-variable regularization of the Fuglede-Kadison determinant.

If this is right

  • The derived SDE system for eigenvalues and overlaps is invariant under simultaneous scale transformations of the left and right eigenvectors.
  • Derivatives of the logarithm of the regularized Fuglede-Kadison determinant recover the time-dependent eigenvalue point process weighted by the diagonal of the overlap matrix.
  • Averaging the SPDEs for the regularized determinant field produces deterministic PDEs satisfied by the expected densities.
  • The auxiliary complex variable is required to regularize the Fuglede-Kadison determinant so that the random field and its variations are well-defined.

Where Pith is reading between the lines

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

  • The scale invariance implies that only the projective directions of the eigenvectors matter for the spectral dynamics, which may simplify numerical sampling of large non-Hermitian spectra.
  • The overlap-weighted point process offers a natural extension of classical Dyson Brownian motion to the non-Hermitian setting that keeps track of eigenvector alignment.
  • The SPDE formulation for the regularized determinant may admit hydrodynamic limits or large-deviation principles when the matrix size grows.

Load-bearing premise

Bi-orthogonality between the right and left eigenvector processes is maintained at every time so that eigenvector rescaling leaves the eigenvalue trajectories invariant.

What would settle it

Numerical integration of the underlying matrix Brownian motion in which the eigenvectors are explicitly rescaled at a chosen time; the eigenvalue paths should remain continuous while the overlap matrix jumps exactly as required by the SDE system.

read the original abstract

The non-Hermitian matrix-valued Brownian motion is the stochastic process of a random matrix whose entries are given by independent complex Brownian motions. The bi-orthogonality relation is imposed between the right and the left eigenvector processes, which allows for their scale transformations with an invariant eigenvalue process. The eigenvector-overlap process is a Hermitian matrix-valued process, each element of which is given by a product of an overlap of right eigenvectors and that of left eigenvectors. We derive a set of stochastic differential equations (SDEs) for the coupled system of the eigenvalue process and the eigenvector-overlap process and prove the scale-transformation invariance of the obtained SDE system. The Fuglede--Kadison (FK) determinant associated with the present matrix-valued process is regularized by introducing an auxiliary complex variable. This variable is necessary to give the stochastic partial differential equations (SPDEs) for the time-dependent random field defined by the regularized FK determinant and for its squared and logarithmic variations. Time-dependent point process of eigenvalues and its variation weighted by the diagonal elements of the eigenvector-overlap process are related to the derivatives of the logarithmic regularized FK-determinant random-field. We also discuss the PDEs obtained by averaging the SPDEs.

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

0 major / 3 minor

Summary. The paper studies the non-Hermitian matrix-valued Brownian motion whose entries are independent complex Brownian motions. Imposing bi-orthogonality between the right and left eigenvector processes (which fixes the gauge so that eigenvalues remain invariant under rescaling), it derives an SDE system coupling the eigenvalue process to the Hermitian eigenvector-overlap process. The manuscript proves that this SDE system is invariant under the residual scale transformations. It then regularizes the Fuglede–Kadison determinant by an auxiliary complex variable, obtains SPDEs for the resulting random field and its squared and logarithmic variations, relates the eigenvalue point process (weighted by diagonal overlaps) to derivatives of the log-regularized determinant, and derives the averaged PDEs.

Significance. If the derivations are correct, the work supplies an explicit Itô-calculus description of the joint eigenvalue–overlap dynamics for non-Hermitian matrix diffusions together with a regularized determinant whose SPDEs encode the weighted point process. These objects are of direct interest in non-Hermitian random-matrix theory and in the study of stochastic operators; the scale-invariance result and the link between the overlap-weighted spectrum and the determinant field are concrete technical contributions.

minor comments (3)
  1. The abstract and introduction state that the SDE system and its invariance are derived and proved, yet the precise statement of the SDE (drift and diffusion coefficients) appears only later; a compact display of the full coupled system early in the paper would improve readability.
  2. Notation for the left and right eigenvector processes, their normalizations, and the overlap matrix is introduced gradually; a single preliminary section collecting all definitions and the bi-orthogonality constraint would reduce forward references.
  3. The regularization of the Fuglede–Kadison determinant is introduced via an auxiliary complex variable z; the precise domain of z and the manner in which the limit z→0 is taken should be stated explicitly when the SPDEs are written.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, recognition of its contributions to non-Hermitian random-matrix theory, and recommendation of minor revision. No specific major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper derives SDEs for the coupled eigenvalue and eigenvector-overlap processes of non-Hermitian matrix Brownian motion via direct application of Itô calculus, imposing bi-orthogonality as an explicit gauge choice that preserves eigenvalues under scale transformations. It then proves invariance of the resulting SDE system under those transformations. No steps reduce by construction to fitted inputs, self-definitions, or load-bearing self-citations; the scale-invariance follows from the derived equations rather than being presupposed. The regularized FK determinant and associated SPDEs are constructed from the same stochastic process without circular renaming or ansatz smuggling. This is a standard stochastic-calculus construction with no internal reduction to its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Central claims rest on the imposed bi-orthogonality assumption and standard background results from stochastic calculus and random matrix theory.

axioms (1)
  • domain assumption Bi-orthogonality relation between right and left eigenvector processes
    Imposed to allow scale transformations while keeping the eigenvalue process invariant.

pith-pipeline@v0.9.0 · 5774 in / 1135 out tokens · 20151 ms · 2026-05-24T08:40:05.220151+00:00 · methodology

discussion (0)

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Works this paper leans on

42 extracted references · 42 canonical work pages

  1. [1]

    Random Ma trices: Theory and Applications 9 (4), 2050015 (2000)

    Akemann, G., Tribe, R., Tsareas, A., Zaboronski, O.: On the deter minantal structure of conditional overlaps for the complex Ginibre ensemble. Random Ma trices: Theory and Applications 9 (4), 2050015 (2000)

    work page 2000

  2. [2]

    D.: Circular law

    Bai, Z. D.: Circular law. Ann. Probab. 25, 494–529 (1997)

    work page 1997

  3. [3]

    A., Speicher, R., Tarnowski, W.: Squared eig envalue condition numbers and eigenvector correlations from the single ring theorem

    Belinschi, S., Nowak, M. A., Speicher, R., Tarnowski, W.: Squared eig envalue condition numbers and eigenvector correlations from the single ring theorem . J. Phys. A: Math. Theor. 50, 105204 (11 pages) (2017)

    work page 2017

  4. [4]

    T., ´Sniady, P., Speicher, R.: Eigenvalues of non-Hermitian random matric es and Brown measure of non-normal operators: Hermitian reductio n and linearization method

    Belinschi, S. T., ´Sniady, P., Speicher, R.: Eigenvalues of non-Hermitian random matric es and Brown measure of non-normal operators: Hermitian reductio n and linearization method. Linear Algebra Appl. 537, 48–83 (2018)

    work page 2018

  5. [5]

    R.: The Cauchy Transform, Potential Theory and Confor mal Mapping

    Bell, S. R.: The Cauchy Transform, Potential Theory and Confor mal Mapping. 2nd ed. CRC Press, Boca Raton, FL (2016)

    work page 2016

  6. [6]

    Bourgade, P., Dubach, G.: The distribution of overlaps between e igenvectors of Ginibre matrices. Probab. Theory Relat. Fields 177, 397–464 (2020)

    work page 2020

  7. [7]

    A., Tarnowski, W., Warcho/suppress l, P.: Dysonian dynamics of the Ginibre ensemble

    Burda, Z., Grela, J., Nowak, M. A., Tarnowski, W., Warcho/suppress l, P.: Dysonian dynamics of the Ginibre ensemble. Phys. Rev. Lett. 113, 104102 (5 pages) (2014)

    work page 2014

  8. [8]

    A., Tarnowski, W., Warcho/suppress l, P.: Unveiling the signifi- cance of eigenvectors in diffusing non-Hermitian matrices by identify ing the underlying Burgers dynamics

    Burda, Z., Grela, J., Nowak, M. A., Tarnowski, W., Warcho/suppress l, P.: Unveiling the signifi- cance of eigenvectors in diffusing non-Hermitian matrices by identify ing the underlying Burgers dynamics. Nucl. Phys. B 897, 421–447 (2015)

    work page 2015

  9. [9]

    J.: Progress on the study of the Ginib re ensemble I: G INUE

    Byun, S.-S., Forrester, P. J.: Progress on the study of the Ginib re ensemble I: G INUE. KIAS Springer Series in Mathematics (to appear), arXiv:math-ph/2211.16223

    work page arXiv

  10. [10]

    J.: Progress on the study of the Gin ibre ensemble II: GINOE and G INSE

    Byun, S.-S., Forrester, P. J.: Progress on the study of the Gin ibre ensemble II: GINOE and G INSE. KIAS Springer Series in Mathematics (to appear), arXiv:math- ph/2301.05022 35

    work page arXiv

  11. [11]

    T., Mehlig, B.: Eigenvector statistics in non-Hermitian random matrix ensembles

    Chalker, J. T., Mehlig, B.: Eigenvector statistics in non-Hermitian random matrix ensembles. Phys. Rev. Lett. 81 (16), 3367–3370 (1998)

    work page 1998

  12. [12]

    K., Hall, B., Kemp, T.: The Brown measure of the free mu ltiplicative Brow- nian motion

    Driver, B. K., Hall, B., Kemp, T.: The Brown measure of the free mu ltiplicative Brow- nian motion. Probab. Theory Relat. Fields 184, 209–273 (2022)

    work page 2022

  13. [13]

    J.: A Brownian-motion model for the eigenvalues of a r andom matrix

    Dyson, F. J.: A Brownian-motion model for the eigenvalues of a r andom matrix. J. Math. Phys. 3, 1191–1198 (1962)

    work page 1962

  14. [14]

    J.: Log-Gases and Random Matrices

    Forrester, P. J.: Log-Gases and Random Matrices. Princeton University Press, Prince- ton, NJ (2010)

    work page 2010

  15. [15]

    V.: Determinant theory in finite factors

    Fuglede, B., Kadison, R. V.: Determinant theory in finite factors . Ann. of Math. 55 (3), 520–530 (1952)

    work page 1952

  16. [16]

    V.: On statistics of bi-orthogonal eigenvectors in real and complex Gini- bre ensembles: Combining partial Schur decomposition with supersy mmetry

    Fyodorov, Y. V.: On statistics of bi-orthogonal eigenvectors in real and complex Gini- bre ensembles: Combining partial Schur decomposition with supersy mmetry. Commun. Math. Phys. 363, 579–603 (2018)

    work page 2018

  17. [17]

    V., Mehlig, B.: Statistics of resonances and nonor thogonal eigenfunctions in a model for single-channel chaotic scattering

    Fyodorov, Y. V., Mehlig, B.: Statistics of resonances and nonor thogonal eigenfunctions in a model for single-channel chaotic scattering. Phys. Rev. E 66, 045202(R) (4 pages) (2002)

    work page 2002

  18. [18]

    V., Osman, M.: Eigenfunction non-orthogonality f actors and the shape of CPA-like dips in a single-channel reflection from lossy chaotic cavit ies

    Fyodorov, Y. V., Osman, M.: Eigenfunction non-orthogonality f actors and the shape of CPA-like dips in a single-channel reflection from lossy chaotic cavit ies. J. Phys. A: Math. Theor. 55, 224013 (22 pages) (2022)

    work page 2022

  19. [19]

    V., Savin, D

    Fyodorov, Y. V., Savin, D. V.: Statistics of resonance width shif ts as a signature of eigenfunction nonorthogonality. Phys. Rev. Lett. 108, 184101 (5 pages) (2012)

    work page 2012

  20. [20]

    V., Tarnowski, W.: Condition numbers for real eige nvalues in the real elliptic Gaussian ensemble

    Fyodorov, Y. V., Tarnowski, W.: Condition numbers for real eige nvalues in the real elliptic Gaussian ensemble. Ann. Henri Poincar´ e 22, 309–330 (2021)

    work page 2021

  21. [21]

    Ginibre, J.: Statistical ensembles of complex, quaternion, and r eal matrices. J. Math. Phys. 6, 440–449 (1965)

    work page 1965

  22. [22]

    L.: Circular law

    Girko, V. L.: Circular law. Theory Probab. Appl. 29(4), 694–706 (1985)

    work page 1985

  23. [23]

    L.: Elliptic law

    Girko, V. L.: Elliptic law. Theory Probab. Appl. 30(4), 677–690 (1986)

    work page 1986

  24. [24]

    Grela J., Warcho/suppress l, P.: Full Dysonian dynamics of the complex Ginibre ensemble. J. Phys. A: Math. Theor. 51, 425203 (26 pages) (2018)

    work page 2018

  25. [25]

    C., Ho, C.-W.: The Brown measure of the sum of a self-adjo int element and an imaginary multiple of a semicircular element

    Hall, B. C., Ho, C.-W.: The Brown measure of the sum of a self-adjo int element and an imaginary multiple of a semicircular element. Lett. Math. Phys. 112, 19 (61 pages) (2022) 36

    work page 2022

  26. [26]

    C., Ho, C.-W.: The heat flow conjecture for random matric es

    Hall, B. C., Ho, C.-W.: The heat flow conjecture for random matric es. arXiv:math.PR/2202.09660

    work page arXiv

  27. [27]

    Ho, C.-W., Zhong, P.: Brown measures of free circular and multiplic ative Brownian motions with self-adjoint and unitary initial conditions. J. Eur. Math . Soc. 25 (6), 2163–2227 (2023)

    work page 2023

  28. [28]

    Ishiwata, S., Kawabi, H., Kotani, M.: Long time asymptotics of non -symmetric random walks on crystal lattices. J. Funct. Anal. 272, 1533–1624 (2017)

    work page 2017

  29. [29]

    A., N¨ orenberg, W., Nowak, M

    Janik, R. A., N¨ orenberg, W., Nowak, M. A., Papp, G., Zahed, I.: C orrelations of eigenvectors for non-Hermitian random-matrix models. Phys. Rev . E 60, 2699–2705 (1999)

    work page 1999

  30. [30]

    Springe r-Verlag, Berlin, Corrected Printing of the Second Edition (1980)

    Kato, T.: Perturbation Theory for Linear Operators. Springe r-Verlag, Berlin, Corrected Printing of the Second Edition (1980)

    work page 1980

  31. [31]

    SpringerBriefs in Mathematical Physic, Vol

    Katori, M.: Bessel Processes, Schramm–Loewner Evolution, a nd the Dyson Model. SpringerBriefs in Mathematical Physic, Vol. 11, Springer, Singapor e (2016)

    work page 2016

  32. [32]

    A., Sommers, H

    Khoruzhenko, B. A., Sommers, H. J.: Non-Hermitian ensembles. In: Akemann, G., Baik, J., Di Francesco, P. (eds.), The Oxford Handbook of Random Matrix Theory, Chapter 18, pp.376–397, Oxford University Press, Oxford (2011 )

    work page 2011

  33. [33]

    T.: Statistical properties of eigenvector s in non-Hermitian Gaus- sian random matrix ensembles: J

    Mehlig, B., Chalker, J. T.: Statistical properties of eigenvector s in non-Hermitian Gaus- sian random matrix ensembles: J. Math. Phys. 41, 3233–3256 (2000)

    work page 2000

  34. [34]

    L.: Random Matrices

    Mehta, M. L.: Random Matrices. 3rd edn. Elsevier, Amsterdam ( 2004)

    work page 2004

  35. [35]

    arXiv:math- ph/2305.13184

    Mezzadri, F., Taylor, H.: A matrix model of a non-Hermitian β -ensemble. arXiv:math- ph/2305.13184

    work page arXiv

  36. [36]

    A., Speicher, R.: Free Probability and Random Matrices

    Mingo, J. A., Speicher, R.: Free Probability and Random Matrices. Fields Institute Monographs 35, Springer, New York (2017)

    work page 2017

  37. [37]

    Ast´ erisque 187–188 (268 pages) (1990)

    Parry, W., Pollicott, M.: Zeta functions and the periodic orbit str ucture of hyperbolic dynamics. Ast´ erisque 187–188 (268 pages) (1990)

    work page 1990

  38. [38]

    J., Crisanti, A., Sompolinsky, H., Stein, Y.: Spectrum of large random asymmetric matrices

    Sommers, H. J., Crisanti, A., Sompolinsky, H., Stein, Y.: Spectrum of large random asymmetric matrices. Phys. Rev. Lett. 60, 1895–1898 (1988)

    work page 1988

  39. [39]

    H.: Random matrices: the circular law

    Tao, T., Vu, V. H.: Random matrices: the circular law. Commun. Co ntemp. Math. 10(2), 261–307 (2008)

    work page 2008

  40. [40]

    N., Embree, M.: Spectra and Pseudospectra: t he Behavior of Nonnormal Matrices and Operators, vol

    Trefethen, L. N., Embree, M.: Spectra and Pseudospectra: t he Behavior of Nonnormal Matrices and Operators, vol. 1. Princeton University Press, Princ eton (2005) 37

    work page 2005

  41. [41]

    Yabuoku, S.: Eigenvalue processes of elliptic Ginibre ensemble and their overlaps. Int. J. Math. Ind. 12(1), 2050003 (17 pages) (2020)

    work page 2020

  42. [42]

    arXiv:math.OA/2108.09844 38

    Zhong, P.: Brown measure of the sum of an elliptic operator and a free random variable in a finite von Neumann algebra. arXiv:math.OA/2108.09844 38

    work page arXiv