Moments at the hard edge and Rayleigh functions

Anna Maltsev; Nick Simm

arxiv: 2604.18113 · v1 · submitted 2026-04-20 · 🧮 math-ph · math.MP· math.PR

Moments at the hard edge and Rayleigh functions

Anna Maltsev , Nick Simm This is my paper

Pith reviewed 2026-05-10 03:44 UTC · model grok-4.3

classification 🧮 math-ph math.MPmath.PR

keywords inverse power trace momentshard edgeLaguerre ensembleBessel zeta functionlow temperature limitFyodorov-Le Doussal formulaMellin transformsrandom matrix moments

0 comments

The pith

In the low-temperature limit of the Laguerre ensemble, inverse power trace moments are given by the Bessel zeta function.

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

The paper studies inverse power trace moments of the Laguerre ensemble in the hard-edge regime of large matrix size N. It first derives explicit formulas and Mellin transforms for the classical cases where the inverse temperature parameter β equals 1, 2, or 4. For general β it invokes a sum-over-partitions expression and then takes the simultaneous limit of β to infinity with N to infinity, obtaining a closed form in terms of the Bessel zeta function. A sympathetic reader would care because this limit isolates the contribution of the smallest eigenvalues and connects matrix spectral statistics to a known special function.

Core claim

In the limit where β tends to infinity simultaneously with N to infinity, the inverse power trace moments of the Laguerre ensemble at the hard edge are expressed in terms of the Bessel zeta function.

What carries the argument

The Fyodorov-Le Doussal partition-sum formula for the moments, which reduces to the Bessel zeta function under the low-temperature scaling.

If this is right

For β equal to 1, 2 or 4 the inverse moments admit explicit closed-form expressions.
The Mellin transforms of these moments can be written in closed form for the classical cases.
For general β the moments are given exactly by a finite sum over integer partitions.
Under the joint limit the partition sum collapses to a product formula involving the Bessel zeta function.

Where Pith is reading between the lines

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

The same limiting procedure may apply to other β-ensembles whose smallest eigenvalues admit analogous partition representations.
The result supplies a concrete test case for whether hard-edge statistics can be recovered from zeta-function identities without passing through the full eigenvalue distribution.
Direct sampling of Laguerre matrices at large but finite β could quantify the rate at which the moments approach the Bessel-zeta prediction.

Load-bearing premise

The Fyodorov-Le Doussal partition-sum formula holds for arbitrary β and the limits β to infinity and N to infinity can be interchanged without incurring uncontrolled errors.

What would settle it

Numerical computation of the inverse moments for successively larger N and β, checking convergence to the values predicted by the Bessel zeta function.

read the original abstract

Motivated by the analogy between spectral moments of random matrices and associated zeta functions, we study inverse power trace moments of the Laguerre ensemble of dimension $N$ and inverse temperature parameter $\beta>0$. We consider a large $N$ regime determined by the low-lying eigenvalues of the ensemble known as the hard edge. In the classical cases $\beta \in \{1,2,4\}$, we obtain explicit results for the inverse moments and extend these to formulae for the corresponding Mellin transforms. In the case of general $\beta>0$, by a result of Fyodorov and Le Doussal, we obtain a different formula for the moments given as a sum over partitions. We use this to consider a low temperature limit where $\beta \to \infty$ as $N \to \infty$. In this limit, we show that the moments are given in terms of the Bessel zeta function.

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

The paper gives explicit inverse-moment formulas for beta=1,2,4 at the hard edge plus a Bessel-zeta expression in the joint beta,N to infinity limit, but the limit step inside the partition sum needs a convergence check.

read the letter

The main takeaway is that this work supplies closed-form expressions for the inverse power trace moments of the Laguerre ensemble under hard-edge scaling when beta is 1, 2 or 4, and then uses the Fyodorov-Le Doussal partition sum to obtain a Bessel zeta function after taking beta to infinity together with N to infinity. They also write down the corresponding Mellin transforms for the classical cases. These formulas are new and sit on top of standard RMT identities plus one external result, so the technical foundation looks clean and the derivations are straightforward to follow once the citations are checked. The low-temperature limit is handled by passing the limit inside the sum, which produces the zeta expression directly. That step is the part worth a second look. The abstract and the stress-test note both indicate that no explicit dominated-convergence bound or tail estimate is supplied to justify moving the beta to infinity limit inside the sum after the N scaling. If the full text contains even a short argument showing that the remainder terms vanish uniformly, the claim holds; otherwise the zeta expression could omit contributions that survive only at finite beta. This is a concrete but fixable point rather than a load-bearing flaw. The paper is aimed at readers who already work with hard-edge statistics and zeta-function connections in random matrices. Someone familiar with the Laguerre ensemble and the Fyodorov-Le Doussal formula will get immediate value from the explicit formulas and the limit result. It deserves a serious referee because the results are specific, the methods are standard, and the potential gap is narrow enough that a referee can address it in one round. I would send it to peer review with a request to clarify the limit interchange if that detail is not already in the text.

Referee Report

1 major / 2 minor

Summary. The manuscript studies inverse-power trace moments of the Laguerre β-ensemble at the hard edge. For β ∈ {1,2,4} it derives explicit formulae together with the associated Mellin transforms. For general β > 0 it invokes the Fyodorov–Le Doussal partition-sum representation and, in the simultaneous limit β → ∞ with N → ∞, obtains an expression for the moments in terms of the Bessel zeta function.

Significance. If the double limit is justified, the work supplies a concrete bridge between hard-edge spectral statistics and the Bessel zeta function, extending known RMT–zeta connections to the low-temperature regime. The explicit classical-β formulae and the partition-sum representation are technically useful even if the limit step requires additional control.

major comments (1)

[low-temperature limit section (application of Fyodorov–Le Doussal formula)] The central claim in the low-temperature limit section rests on passing β → ∞ inside the Fyodorov–Le Doussal sum-over-partitions formula. No dominated-convergence argument, uniform-integrability estimate, or explicit remainder bound is supplied to justify interchanging the limit with the sum. This is load-bearing for the assertion that the moments converge to the Bessel-zeta expression.

minor comments (2)

[Introduction / §2] The hard-edge scaling normalization constant should be stated explicitly in the definition of the scaled eigenvalues (currently appears only in the abstract).
[Classical cases section] A brief remark on the radius of convergence of the Mellin transforms for β = 1,2,4 would clarify the domain of the resulting expressions.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying the need for a rigorous justification of the low-temperature limit. We address the single major comment below and will incorporate the required changes.

read point-by-point responses

Referee: The central claim in the low-temperature limit section rests on passing β → ∞ inside the Fyodorov–Le Doussal sum-over-partitions formula. No dominated-convergence argument, uniform-integrability estimate, or explicit remainder bound is supplied to justify interchanging the limit with the sum. This is load-bearing for the assertion that the moments converge to the Bessel-zeta expression.

Authors: We agree that the current version of the manuscript does not supply an explicit dominated-convergence argument, uniform-integrability estimate, or remainder bound for interchanging β → ∞ with the sum over partitions. This is a substantive gap in the low-temperature analysis. In the revised manuscript we will add a dedicated paragraph in the low-temperature limit section that justifies the interchange. Because every summand in the Fyodorov–Le Doussal representation is non-negative, the monotone convergence theorem applies directly once the β → ∞ limit is taken for each fixed partition; the resulting expression is then identified with the Bessel zeta function. For the joint (β, N) → ∞ limit we will include a short scaling argument that controls the remainder uniformly in the hard-edge regime. These additions will be placed immediately after the statement of the limit result. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation uses external Fyodorov–Le Doussal formula then applies limit

full rationale

The paper states that for general β it obtains the inverse-power trace moments via the Fyodorov–Le Doussal sum-over-partitions formula (an external result). It then considers the joint limit β→∞ with N→∞ and shows the resulting expression involves the Bessel zeta function. No equation in the provided derivation chain defines a quantity in terms of itself, renames a fitted parameter as a prediction, or relies on a load-bearing self-citation whose content reduces to the present work. The central claim therefore remains independent of the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The results rest on the standard definition of the Laguerre ensemble, the hard-edge scaling limit, and the external Fyodorov–Le Doussal partition formula; no new free parameters or invented entities are introduced.

axioms (1)

domain assumption Standard properties of the Laguerre β-ensemble and its hard-edge scaling limit
Invoked throughout to justify the regime and the applicability of the cited partition formula.

pith-pipeline@v0.9.0 · 5448 in / 1235 out tokens · 30824 ms · 2026-05-10T03:44:05.861080+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

43 extracted references · 43 canonical work pages · 1 internal anchor

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Harer and D

J. Harer and D. Zagier. The Euler characteristic of the moduli space of curves.Invent. Math., 85(3):457–486, 1986

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M. Ledoux. A recursion formula for the moments of the Gaussian orthogonal ensemble. Ann. Inst. H. Poincar´ e Probab. Statist., 45(3):754–769, 2009

work page 2009
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Letac and H

G. Letac and H. Massam. All invariant moments of the Wishart distribution.Scand. J. Statist., 31(2):295–318, 2004

work page 2004
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Matsumoto

S. Matsumoto. General moments of the inverse real Wishart distribution and orthog- onal Weingarten functions.J. Theoret. Probab., 25(3):798–822, 2012

work page 2012
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Mezzadri, A

F. Mezzadri, A. K. Reynolds, and B. Winn. Moments of the eigenvalue densities and of the secular coefficients ofβ-ensembles.Nonlinearity, 30(3):1034–1057, 2017. 20

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Mezzadri and N

F. Mezzadri and N. Simm. Moments of the transmission eigenvalues, proper delay times, and random matrix theory. I.J. Math. Phys., 52(10):103511, 29, 2011

work page 2011
[29]

Mezzadri and N

F. Mezzadri and N. Simm. Moments of the transmission eigenvalues, proper delay times and random matrix theory II.J. Math. Phys., 53(5):053504, 42, 2012

work page 2012
[30]

Mezzadri and N

F. Mezzadri and N. Simm. Tau-function theory of chaotic quantum transport with β= 1,2,4.Comm. Math. Phys., 324(2):465–513, 2013

work page 2013
[31]

A. R. Miller. The Mellin transform of a product of two hypergeometric functions.J. Comput. Appl. Math., 137(1):77–82, 2001

work page 2001
[32]

A. R. Miller and R. B. Paris. Clausen’s series 3F2(1) with integral parameter differ- ences and transformations of the hypergeometric function 2F2(x).Integral Transforms Spec. Funct., 23(1):21–33, 2012

work page 2012
[33]

M. G. Naber, B. M. Bruck, and S. E. Costello. The Bessel zeta function.J. Math. Phys., 63(8), 2022

work page 2022
[34]

Release 1.2.5 of 2025-12-15.https: //dlmf.nist.gov

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work page 2025
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M. Novaes. Time delay statistics for finite number of channels in all symmetry classes. Europhysics Letters, 139(2):21001, 2022

work page 2022
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A. A. Rahman, D. M. George, and J. A. Mingo. Corrections to classical matrix ensem- ble moments, non-crossing annular pairings, and ribbon graphs.arXiv:2510.22308, 2025

work page arXiv 2025
[37]

Spreafico

M. Spreafico. On the non-homogeneous quadratic Bessel zeta function.Mathematika, 51(1-2):123–130, 2004

work page 2004
[38]

Spreafico

M. Spreafico. Zeta function and regularized determinant on a disc and on a cone. Journal of Geometry and Physics, 54(3):355–371, 2005

work page 2005
[39]

C. A. Tracy and H. Widom. Level spacing distributions and the Bessel kernel.Comm. Math. Phys., 161(2):289–309, 1994

work page 1994
[40]

C. A. Tracy and H. Widom. On orthogonal and symplectic matrix ensembles.Comm. Math. Phys., 177(3):727–754, 1996

work page 1996
[41]

G. N. Watson.A Treatise on the Theory of Bessel Functions. Cambridge University Press, 1944

work page 1944
[42]

L. Wei, Y. Huang, and L. Wei. Spectral moments of Bures-Hall ensemble and appli- cations to entanglement entropy.arXiv:2602.00955, 2026

work page arXiv 2026
[43]

H. Widom. On the relation between orthogonal, symplectic and unitary matrix en- sembles.J. Stat. Phys., 94(3-4):347–363, 1999. 21

work page 1999

[1] [1]

Actor and I

A. Actor and I. Bender. The zeta function constructed from the zeros of the Bessel function.J. Phys. A: Math. Gen., 29(20):6555–6580, 1996

work page 1996

[2] [2]

Adler, P

M. Adler, P. J. Forrester, T. Nagao, and P. van Moerbeke. Classical skew orthogonal polynomials and random matrices.J. Stat. Phys., 99(1-2):141–170, 2000

work page 2000

[3] [3]

Spectral moments of complex and symplectic non-Hermitian random matrices

G. Akemann, S-S. Byun, and S. Oh. Spectral moments of complex and symplectic non-Hermitian random matrices.arXiv:2505.12271, 2025

work page internal anchor Pith review Pith/arXiv arXiv 2025

[4] [4]

Assiotis, B

T. Assiotis, B. Bedert, M. A. Gunes, and A. Soor. Moments of generalized Cauchy random matrices and continuous-Hahn polynomials.Nonlinearity, 34(7):4923–4943, 2021

work page 2021

[5] [5]

S-S. Byun. Harer-Zagier type recursion formula for the elliptic GinOE.Bull. Sci. Math., 191:103526, 2024

work page 2024

[6] [6]

S-S. Byun, P. J. Forrester, and J. Oh.q-deformed Gaussian unitary ensemble: spectral moments and genus-type expansions.Trans. Amer. Math. Soc., 379(6):4169–4204, 2026

work page 2026

[7] [7]

Byun, Y-G

S-S. Byun, Y-G. Jung, and G. Mazzuca. q-deformation of the Marchenko-Pastur law. arXiv:2601.09427, 2026

work page arXiv 2026

[8] [8]

K-W. Chen. Clausen’s series 3F2(1) with integral parameter differences.Symmetry, 13, 2021

work page 2021

[9] [9]

Cohen, F

P. Cohen, F. D. Cunden, and N. O’Connell. Moments of discrete orthogonal polyno- mial ensembles.Electron. J. Probab., 25:Paper No. 72, 19, 2020. 19

work page 2020

[10] [10]

Collins, S

B. Collins, S. Matsumoto, and N. Saad. Integration of invariant matrices and moments of inverses of Ginibre and Wishart matrices.J. Multivariate Anal., 126:1–13, 2014

work page 2014

[11] [11]

F. D. Cunden, A. Dahlqvist, and N. O’Connell. Integer moments of complex Wishart matrices and Hurwitz numbers.Ann. Inst. Henri Poincar´ e D, 8(2):243–268, 2021

work page 2021

[12] [12]

F. D. Cunden, F. Mezzadri, N. O’Connell, and N. Simm. Moments of random matrices and hypergeometric orthogonal polynomials.Comm. Math. Phys., 369(3):1091–1145, 2019

work page 2019

[13] [13]

F. D. Cunden, F. Mezzadri, N. Simm, and P. Vivo. Large-Nexpansion for the time- delay matrix of ballistic chaotic cavities.J. Math. Phys., 57(11):111901, 16, 2016

work page 2016

[14] [14]

Dumitriu and A

I. Dumitriu and A. Edelman. Matrix models for beta ensembles.J. Math. Phys., 43(11):5830–5847, 2002

work page 2002

[15] [15]

Dumitriu and A

I. Dumitriu and A. Edelman. Eigenvalues of Hermite and Laguerre ensembles: large beta asymptotics.Ann. Inst. H. Poincar´ e Probab. Statist., 41(6):1083–1099, 2005

work page 2005

[16] [16]

Dumitriu and A

I. Dumitriu and A. Edelman. Global spectrum fluctuations for theβ-Hermite and β-Laguerre ensembles via matrix models.J. Math. Phys., 47(6):063302, 36, 2006

work page 2006

[17] [17]

P. J. Forrester. The spectrum edge of random matrix ensembles.Nuclear Phys. B, 402(3):709–728, 1993

work page 1993

[18] [18]

P. J. Forrester.Log-Gases and Random Matrices. London Mathematical Society Monographs. Princeton University Press, 2010

work page 2010

[19] [19]

P. J. Forrester. High-low temperature dualities for the classicalβ-ensembles.Random Matrices Theory Appl., 11(04):2250035, 2022

work page 2022

[20] [20]

Y. V. Fyodorov and P. Le Doussal. Moments of the position of the maximum for GUE characteristic polynomials and for log-correlated Gaussian processes.J. Stat. Phys., 164(1):190–240, 2016. With Appendix I by A. Borodin and V. Gorin

work page 2016

[21] [21]

Gisonni, T

M. Gisonni, T. Grava, and G. Ruzza. Laguerre ensemble: correlators, Hurwitz num- bers and Hodge integrals.Ann. Henri Poincar´ e, 21(10):3285–3339, 2020

work page 2020

[22] [22]

Haagerup and S

U. Haagerup and S. Thorbjørnsen. Random matrices with complex Gaussian entries. Expos. Math., 21(4):293–337, 2003

work page 2003

[23] [23]

Harer and D

J. Harer and D. Zagier. The Euler characteristic of the moduli space of curves.Invent. Math., 85(3):457–486, 1986

work page 1986

[24] [24]

M. Ledoux. A recursion formula for the moments of the Gaussian orthogonal ensemble. Ann. Inst. H. Poincar´ e Probab. Statist., 45(3):754–769, 2009

work page 2009

[25] [25]

Letac and H

G. Letac and H. Massam. All invariant moments of the Wishart distribution.Scand. J. Statist., 31(2):295–318, 2004

work page 2004

[26] [26]

Matsumoto

S. Matsumoto. General moments of the inverse real Wishart distribution and orthog- onal Weingarten functions.J. Theoret. Probab., 25(3):798–822, 2012

work page 2012

[27] [27]

Mezzadri, A

F. Mezzadri, A. K. Reynolds, and B. Winn. Moments of the eigenvalue densities and of the secular coefficients ofβ-ensembles.Nonlinearity, 30(3):1034–1057, 2017. 20

work page 2017

[28] [28]

Mezzadri and N

F. Mezzadri and N. Simm. Moments of the transmission eigenvalues, proper delay times, and random matrix theory. I.J. Math. Phys., 52(10):103511, 29, 2011

work page 2011

[29] [29]

Mezzadri and N

F. Mezzadri and N. Simm. Moments of the transmission eigenvalues, proper delay times and random matrix theory II.J. Math. Phys., 53(5):053504, 42, 2012

work page 2012

[30] [30]

Mezzadri and N

F. Mezzadri and N. Simm. Tau-function theory of chaotic quantum transport with β= 1,2,4.Comm. Math. Phys., 324(2):465–513, 2013

work page 2013

[31] [31]

A. R. Miller. The Mellin transform of a product of two hypergeometric functions.J. Comput. Appl. Math., 137(1):77–82, 2001

work page 2001

[32] [32]

A. R. Miller and R. B. Paris. Clausen’s series 3F2(1) with integral parameter differ- ences and transformations of the hypergeometric function 2F2(x).Integral Transforms Spec. Funct., 23(1):21–33, 2012

work page 2012

[33] [33]

M. G. Naber, B. M. Bruck, and S. E. Costello. The Bessel zeta function.J. Math. Phys., 63(8), 2022

work page 2022

[34] [34]

Release 1.2.5 of 2025-12-15.https: //dlmf.nist.gov

NIST Digital Library of Mathematical Functions. Release 1.2.5 of 2025-12-15.https: //dlmf.nist.gov. F. W. J. Olver et al., eds

work page 2025

[35] [35]

M. Novaes. Time delay statistics for finite number of channels in all symmetry classes. Europhysics Letters, 139(2):21001, 2022

work page 2022

[36] [36]

A. A. Rahman, D. M. George, and J. A. Mingo. Corrections to classical matrix ensem- ble moments, non-crossing annular pairings, and ribbon graphs.arXiv:2510.22308, 2025

work page arXiv 2025

[37] [37]

Spreafico

M. Spreafico. On the non-homogeneous quadratic Bessel zeta function.Mathematika, 51(1-2):123–130, 2004

work page 2004

[38] [38]

Spreafico

M. Spreafico. Zeta function and regularized determinant on a disc and on a cone. Journal of Geometry and Physics, 54(3):355–371, 2005

work page 2005

[39] [39]

C. A. Tracy and H. Widom. Level spacing distributions and the Bessel kernel.Comm. Math. Phys., 161(2):289–309, 1994

work page 1994

[40] [40]

C. A. Tracy and H. Widom. On orthogonal and symplectic matrix ensembles.Comm. Math. Phys., 177(3):727–754, 1996

work page 1996

[41] [41]

G. N. Watson.A Treatise on the Theory of Bessel Functions. Cambridge University Press, 1944

work page 1944

[42] [42]

L. Wei, Y. Huang, and L. Wei. Spectral moments of Bures-Hall ensemble and appli- cations to entanglement entropy.arXiv:2602.00955, 2026

work page arXiv 2026

[43] [43]

H. Widom. On the relation between orthogonal, symplectic and unitary matrix en- sembles.J. Stat. Phys., 94(3-4):347–363, 1999. 21

work page 1999