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arxiv: 2606.11958 · v1 · pith:73GSZFFVnew · submitted 2026-06-10 · 🧮 math-ph · math.MP

On determinantal formulas for hermitian random matrices

Di Yang , Jiayi Zhao , Jian Zhou This is my paper

Pith reviewed 2026-06-27 08:12 UTC · model grok-4.3

classification 🧮 math-ph math.MP
keywords hermitian matrix modelsdeterminantal formulasconnected k-point functionsKP integrabilityaffine coordinatesdualityrandom matrices
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The pith

Hermitian matrix models have determinantal formulas for connected k-point functions, shown by direct proof.

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

The paper establishes determinantal expressions for the connected k-point correlation functions of hermitian matrix models through a direct algebraic argument. It supplies independent proofs that these models are integrable in the KP sense and that their affine coordinates admit explicit formulas derived from the KP hierarchy. Duality relations are also shown to hold for particular choices of the models. A reader would care because these formulas organize the statistical correlations of eigenvalues in a compact, computable way and tie random-matrix ensembles to the theory of integrable hierarchies.

Core claim

We give a direct proof of determinantal formulas for connected k-point functions for hermitian matrix models. We also give a new proof of KP integrability for them. From the viewpoint of KP hierarchy, we further give a new proof of the explicit formula for the corresponding affine coordinates. Furthermore, duality for some hermitian matrix models is proved.

What carries the argument

Direct algebraic derivation of determinantal formulas for connected k-point functions, supplemented by KP-hierarchy identities.

If this is right

  • Connected eigenvalue correlations are completely determined by a determinant built from a kernel.
  • The partition functions and correlation functions satisfy the KP hierarchy equations.
  • Affine coordinates on the Sato Grassmannian are given by explicit residue formulas.
  • Certain pairs of hermitian matrix models are related by an explicit duality map.

Where Pith is reading between the lines

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

  • The same direct-proof strategy could be tested on non-hermitian or beta-ensembles to see whether the determinantal property survives.
  • The explicit affine-coordinate formulas might simplify numerical sampling of large-N eigenvalue distributions.
  • Duality statements could be used to relate correlation functions across different potentials without recomputing the full partition function.

Load-bearing premise

The algebraic manipulations and combinatorial identities invoked in the direct proof are assumed to hold without gaps for the full range of hermitian matrix models considered.

What would settle it

An explicit computation of the connected two-point function in the Gaussian unitary ensemble that fails to match the claimed determinantal expression.

read the original abstract

In this paper, we give a direct proof of determinantal formulas for connected $k$-point functions for hermitian matrix models. We also give a new proof of KP integrability for them. From the viewpoint of KP hierarchy, we further give a new proof of the explicit formula for the corresponding affine coordinates. Furthermore, duality for some hermitian matrix models is proved.

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 / 2 minor

Summary. The paper gives a direct proof of determinantal formulas for connected k-point functions for hermitian matrix models. It also gives a new proof of KP integrability for them. From the viewpoint of the KP hierarchy, it gives a new proof of the explicit formula for the corresponding affine coordinates. Furthermore, duality for some hermitian matrix models is proved.

Significance. If the direct proofs hold without gaps, the work supplies streamlined derivations of these formulas that avoid indirect routes, strengthening the link between hermitian matrix models and the KP hierarchy. No machine-checked proofs or reproducible code are mentioned, but the explicit combinatorial and algebraic approach would be a strength if the identities are verified for the full class of models.

minor comments (2)
  1. [Abstract] The abstract states that direct proofs are given, but without section numbers or equation references in the summary, it is hard to locate the key algebraic steps or combinatorial identities used in the derivations.
  2. The scope of 'some hermitian matrix models' for the duality result should be stated more precisely, e.g., by listing the specific potentials or matrix sizes to which it applies.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and the recommendation of minor revision. The referee summary correctly reflects the paper's contributions on direct proofs of determinantal formulas, KP integrability, affine coordinates, and duality.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper presents direct algebraic proofs of determinantal formulas for connected k-point functions, KP integrability, and affine coordinates for hermitian matrix models, starting from the KP hierarchy viewpoint. No steps reduce by construction to fitted parameters, self-definitions, or load-bearing self-citations; the claimed derivations rely on combinatorial identities and manipulations treated as independent within the stated scope. The central results are therefore self-contained against external mathematical benchmarks rather than circular.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no explicit free parameters, axioms, or invented entities; full text would be required to audit the algebraic assumptions underlying the direct proofs.

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discussion (0)

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Reference graph

Works this paper leans on

62 extracted references · 7 linked inside Pith

  1. [1]

    Adler M., van Moerbeke P. (1995). Matrix integrals, Toda symmetries, Virasoro constraints, and orthogonal polynomials. Duke Math. J.80, 863–911

    1995

  2. [2]

    Alexandrov A., Bychkov B., Dunin-Barkowski P., Kazarian M., Shadrin S. (2025). KP integrability through thex−yswap relation. Sel. Math. New Ser.31, 42

    2025

  3. [3]

    Balogh F., Yang D. (2017). Geometric interpretation of Zhou’s explicit formula for the Witten– Kontsevich tau function. Lett. Math. Phys.107, 1837–1857

    2017

  4. [4]

    Fibonacci numbers and orthogonal polynomials

    Berg C. Fibonacci numbers and orthogonal polynomials. arXiv:math/0609283

    Pith/arXiv arXiv

  5. [5]

    Bergère M. (2006). Correlation functions of complex matrix models. J. Phys. A: Math. Gen.39, 8749–8773

    2006

  6. [6]

    Determinantal formulae and loop equations

    Bergère M., Eynard B. Determinantal formulae and loop equations. arXiv:0901.3273v2

    Pith/arXiv arXiv

  7. [7]

    Bertola M., Dubrovin B., Yang D. (2016). Correlation functions of the KdV hierarchy and applica- tions to intersection numbers overMg,n. Phys. D.327, 30–57

    2016

  8. [8]

    Bertola M., Dubrovin B., Yang D. (2021). Simple Lie algebras, Drinfeld–Sokolov hierarchies, and multi-point correlation functions. Mosc. Math. J.21, 233–270

    2021

  9. [9]

    BessisD., ItzyksonC., ZuberJ.B.(1980).Quantumfieldtheorytechniquesingraphicalenumeration. Adv. Appl. Math.1, 109–157

    1980

  10. [10]

    Brézin E., Itzykson C., Parisi G., Zuber J.B. (1978). Planar diagrams. Comm. Math. Phys.59, 35–51

    1978

  11. [11]

    Christoffel E.B. (1858). Über die Gaußische Quadratur und eine Verallgemeinerung derselben. J. Reine Angew. Math.55, 61–82

  12. [12]

    Darboux G. (1878). Mémoire sur l’approximation des fonctions de très-grands nombres, et sur une classe étendue de développements en série, J. Math. Pures Appl.4, 5–56

  13. [13]

    (2000).Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach(Vol

    Deift P. (2000).Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach(Vol. 3). American Mathematical Soc

    2000

  14. [14]

    Dubrovin B., Yang D. (2017). Generating series for GUE correlators. Lett. Math. Phys.107, 1971– 2012

    2017

  15. [15]

    Dubrovin B., Yang D., Zagier D. (2020). Gromov–Witten invariants of the Riemann sphere. Pure Appl. Math. Q.16, 153–190

    2020

  16. [16]

    Dubrovin B., Yang D., Zagier D. (2021). On tau-functions for the KdV hierarchy. Selecta Math.27, 47 pp

    2021

  17. [17]

    Dyson F.J. (1962). Statistical theory of energy levels of complex systems III. J. Math. Phys.3, 166–175

    1962

  18. [18]

    Dyson F.J. (1970). Correlations between the eigenvalues of a random matrix. Comm. Math. Phys. 19, 235–250

    1970

  19. [19]

    arXiv:1510.04430v2

    Eynard B., Kimura T., Ribault S.Random matrices. arXiv:1510.04430v2

    Pith/arXiv arXiv

  20. [20]

    Forrester P. (1993). The spectrum edge of random matrix ensembles. Nucl. Phys. B.402, 709–728. 23

    1993

  21. [21]

    Fu A., Yang D. (2022). The matrix-resolvent method to tau-functions for the nonlinear Schrödinger hierarchy. J. Geom. Phys.179, 104592

    2022

  22. [22]

    (1980).The Theory of Matrices

    Gantmacher F.R. (1980).The Theory of Matrices. Volume 1. Chelsea Publishing Company, New York

    1980

  23. [23]

    Gerasimov A., Marshakov A., Mironov A., Morozov A., Orlov A. (1991). Matrix models of two- dimensional gravity and Toda theory. Nucl. Phys. B.357, 565–618

    1991

  24. [24]

    Gisonni M., Grava T., Ruzza G. (2020). Laguerre Ensemble: Correlators, Hurwitz Numbers and Hodge Integrals. Ann. Henri Poincaré21, 3285–3339

    2020

  25. [25]

    Gisonni M., Grava T., Ruzza G. (2021). Jacobi ensemble, Hurwitz numbers and Wilson polynomials. Lett. Math. Phys.111, 67

    2021

  26. [26]

    Gisonni M., Grava T., Ruzza G. (2023). Correction to: Jacobi ensemble, Hurwitz numbers and Wilson polynomials. Lett. Math. Phys.113, 86

    2023

  27. [27]

    Haine L., Horozov E. (1993). Toda Orbits of Laguerre-polynomials and Representations of the Virasoro Algebra. Bull. Sci. Math.117, 485–518

    1993

  28. [28]

    Haine L., Semengue J. (1999). The Jacobi polynomial ensemble and the Painlevé VI equation. J. Math. Phys.40, 2117–2134

    1999

  29. [29]

    Harer J., Zagier D. (1986). The Euler characteristic of the moduli space of curves. Invent Math85, 457–485

    1986

  30. [30]

    Harnad J., Ènolskii V.Z. (2011). Schur function expansions of KPτ-functions associated to algebraic curves. Russ. Math. Surv.66, 767–807

    2011

  31. [31]

    (1984).Algebraic Methods for Toeplitz-like Matrices and Operators

    Heinig G., Rost K. (1984).Algebraic Methods for Toeplitz-like Matrices and Operators. Birkhäuser Basel, Basel

    1984

  32. [32]

    ’t Hooft G. (1974). A planar diagram theory for strong interactions. Nucl. Phys. B.72, 461–473

    1974

  33. [33]

    ’t Hooft G. (1974). A two-dimensional model for mesons. Nucl. Phys. B.75, 461–470

    1974

  34. [34]

    Jacobi C. (1859). Untersuchungen über die Differentialgleichung der hypergeometrischen Reihe. J. Reine Angew. Math.56, 149–165

  35. [35]

    Kaneko M., Zagier D. (1998). Supersingular j-invariants, hypergeometric series, and Atkin’s orthog- onal polynomials. AMS/IP Stud. in Adv. Math.7, 97–126

    1998

  36. [36]

    Kerov S.V. (1993). Transition probabilities for continual young diagrams and the Markov moment problem. Funct. Anal. Its Appl.27, 104–117

    1993

  37. [37]

    König W. (2005). Orthogonal polynomial ensembles in probability theory. Probab. Surv.2, 385–447

    2005

  38. [38]

    Laguerre E. (1879). Sur l’intégrale R ∞ x e−xdx x . Bull. Soc. Math. France.7, 72–81

  39. [39]

    (1998).Symmetric Functions and Hall Polynomials

    Macdonald I.G. (1998).Symmetric Functions and Hall Polynomials. 2nd Edition. Clarendon Press

    1998

  40. [40]

    Martinec E.J. (1991). On the origin of integrability in matrix models. Commun. Math. Phys.138, 437–449

    1991

  41. [41]

    (2004).Random Matrices

    Mehta M.L. (2004).Random Matrices. 3rd Edition. Academic Press, London

    2004

  42. [42]

    Morozov A., Shakirov S. (2009). Exact 2-point function in Hermitian matrix model. J. High Energy Phys.2009, 003

    2009

  43. [43]

    Plancherel, M., Rotach, W. (1929). Sur les valeurs asymptotiques des polynomes d’HermiteHn(x) = (−I) ne x2 2 dn dxn e− x2 2 . Comment. Math. Helv.1, 227–254

    1929

  44. [44]

    Rota G.C. (1964). On the foundations of combinatorial theory I. Theory of Möbius Functions. Z. Wahrscheinlichkeitstheorie verw Gebiete2. 340–368. 24

    1964

  45. [45]

    Ruzza G. (2019). Tau functions: theory and applications to matrix models and enumerative geom- etry. Doctoral dissertation, SISSA

    2019

  46. [46]

    North-Holland Math

    SatoM.(1983).SolitonequationsasdynamicalsystemsoninfinitedimensionalGrassmannmanifold. North-Holland Math. Stud.81, 259–271

    1983

  47. [47]

    Shaw J.C., Tu M.H., Yen H.C. (1992). Matrix Models at Finite N and the KP Hierarchy. Chinese Journ. Phys.30, 497-507

    1992

  48. [48]

    Sonine N. (1880). Recherches sur les fonctions cylindriques et le développement des fonctions con- tinues en séries. Math. Ann.16, 1–80

  49. [49]

    Soshnikov A. (2000). Determinantal random point fields. Russian Math. Surveys.55, 923–975

    2000

  50. [50]

    (1975).Orthogonal Polynomials

    Szegö G. (1975).Orthogonal Polynomials. 4th Edition. AMS

    1975

  51. [51]

    Yang D. (2020). On tau-functions for the Toda lattice hierarchy. Lett. Math. Phys.110, 555–583

    2020

  52. [52]

    Yang D. (2024). GUE via Frobenius manifolds. I. From matrix gravity to topological gravity and back. Acta Math. Sin. Engl. Ser.40, 383–405

    2024

  53. [53]

    Yang D., Zhou J. (2023). Grothendieck’s dessins d’enfants in a web of dualities. III. J. Phys. A: Math. Theor.56, 055–201

    2023

  54. [54]

    Yang D., Zhou J. (2025). From Toda hierarchy to KP hierarchy. SIGMA.21, 068

    2025

  55. [55]

    Yin S. (2022). The Generalized Hermitian Matrix Model and Examples from Statistical Physics. Doctoral dissertation, Tsinghua University

    2022

  56. [56]

    Explicit Formula for Witten-Kontsevich Tau-Function

    Zhou J. Explicit Formula for Witten-Kontsevich Tau-Function. arXiv:1306.5429

    Pith/arXiv arXiv

  57. [57]

    Emergent Geometry and Mirror Symmetry of A Point

    Zhou J. Emergent Geometry and Mirror Symmetry of A Point. arXiv:1507.01679v1

    Pith/arXiv arXiv

  58. [58]

    Hermitian One-Matrix Model and KP Hierarchy

    Zhou J. Hermitian One-Matrix Model and KP Hierarchy. arXiv:1809.07951v1

    Pith/arXiv arXiv

  59. [59]

    Grothendieck’s dessins d’enfants in a web of dualities

    Zhou J. Grothendieck’s dessins d’enfants in a web of dualities. arXiv:1905.10773

    Pith/arXiv arXiv 1905

  60. [60]

    Integrable Hierarchies and Mirror Symmetry in Probability Theory and Combinatorics

    Zhou J. Integrable Hierarchies and Mirror Symmetry in Probability Theory and Combinatorics. I. Tau-Functions from Probability Theory and Lattice Path Counting.Preprint

  61. [61]

    Integrable Hierarchies and Mirror Symmetry in Probability Theory and Combinatorics

    Zhou J. Integrable Hierarchies and Mirror Symmetry in Probability Theory and Combinatorics. II. Mirror Symmetry for Dyck Paths and Motzkin Paths.Preprint

  62. [62]

    Integrable Hierarchies and Mirror Symmetry in Probability Theory and Combinatorics

    Zhou J. Integrable Hierarchies and Mirror Symmetry in Probability Theory and Combinatorics. III. Mirror Symmetry for Schröder Paths and Delannoy Paths.Preprint. 25