arxiv: 2604.25078 · v1 · submitted 2026-04-28 · 🧮 math-ph · math.MP

Recognition: unknown

Gegenbauer polynomials and fluctuation properties of the one-dimensional Riesz gas

Peter J. Forrester

Authors on Pith no claims yet

Pith reviewed 2026-05-07 14:54 UTC · model grok-4.3

classification 🧮 math-ph math.MP

keywords Riesz gasGegenbauer polynomialslinear statisticscovariancefluctuationsone-dimensional gasespower-law interactionsinfinite density limit

0 comments

The pith

The covariance of smooth linear statistics for the one-dimensional Riesz gas equals a sum over Gegenbauer polynomial components for interaction exponents s in (-1,1).

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

The paper applies a functional derivative technique to the Riesz gas of particles on the interval [-1,1] in the infinite density limit. It produces an explicit formula for the covariance of any two smooth linear statistics when the pair potential is |x-x'|^{-s} with s between -1 and 1 but not zero. The formula is written as a sum over n of the Fourier coefficients of the two statistics against the Gegenbauer polynomials C_n^{(s/2)}(x), recovering the familiar cosine series when s approaches zero. In the special case of power-sum statistics the sum collapses to a closed product of gamma functions that the authors compare with independent exact results.

Core claim

In the infinite density limit with particle support restricted to [-1,1], the covariance of two smooth linear statistics for the Riesz gas with exponent s in (-1,1) and s not zero is given by a sum over Fourier components of the linear statistics with respect to the Gegenbauer polynomial basis {C_n^{(s/2)}(x)}, which reduces to a product of gamma functions for power-sum statistics.

What carries the argument

The functional derivative method applied to the Riesz pair potential, which converts the covariance into an expansion in the Gegenbauer polynomials C_n^{(s/2)}(x) of degree s/2.

Load-bearing premise

The infinite density limit with particle support restricted to the interval [-1,1] allows the functional derivative method to produce the stated covariance formula for s in (-1,1) excluding zero.

What would settle it

Numerical sampling of many particles on [-1,1] with Riesz repulsion of strength s=0.5, computation of the empirical covariance between two chosen smooth linear statistics such as the second and third power sums, and direct numerical check against the gamma-function product predicted by the formula.

read the original abstract

The Riesz gas in one-dimension consists of particles interacting via a pair potential, ${\rm sgn}(s) |x - x'|^{-s}$, $s \ne 0$ and $-\log | x - x'|$ for $s=0$. In the infinite density limit, with the particle support the interval $[-1,1]$, we apply a functional derivative method due to Beenakker to compute the covariance of two smooth linear statistics for the Riesz gas with exponent $s \in (-1,1)$, $s \ne 0$. This we give in terms of a sum over Fourier components of the linear statistics with respect to a Gegenbauer polynomial $\{C_n^{(s/2)}(x) \}$ basis, which generalises a known form in the case $s=0$ involving a cosine expansion. For the power sum linear statistic, our general formula can be reduced to a product of gamma function form, and compared against recent exact results in the literature for this case.

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

Forrester supplies an explicit Gegenbauer-polynomial formula for the covariance of linear statistics in the 1D Riesz gas at general s, which reduces to gamma functions for power sums and matches known results.

read the letter

The paper delivers a concrete extension of the covariance formula for smooth linear statistics in the one-dimensional Riesz gas. Using Beenakker's functional derivative method in the infinite-density limit on [-1,1], it writes the covariance as a sum over coefficients in the Gegenbauer basis C_n^{(s/2)}(x) for s in (-1,1) excluding zero. This directly generalizes the cosine expansion that holds for the logarithmic case s=0. For power-sum statistics the expression collapses to a product of gamma functions, and the author notes that this matches independent exact results already in the literature. That match provides a useful cross-check without relying on self-fitting. The equilibrium density (1-x^2)^{(s-1)/2} is precisely the weight that makes the Gegenbauer polynomials orthogonal and eigenfunctions of the Riesz integral operator, so the covariance comes out diagonal by construction. This is the cleanest part of the work. The derivation stays strictly inside the infinite-density regime on a fixed interval, so it does not cover finite-N corrections or how the fluctuations scale when the support is allowed to vary. The abstract gives no error estimates or convergence details for the functional derivative step, though the external grounding in Beenakker's technique and the gamma-function check reduce the risk of hidden circularity. No internal contradictions appear in the stated construction. This is a paper for readers already working on exact results for 1D log-gases and their Riesz generalizations, or on fluctuation formulas in random-matrix ensembles with power-law kernels. Someone who knows the s=0 case and the functional-derivative approach will extract the new formula quickly and can apply it to further statistics. It is worth sending to peer review because the explicit basis expansion and the gamma reduction are verifiable advances that sit on top of established methods rather than replacing them.

Referee Report

0 major / 3 minor

Summary. The manuscript applies Beenakker's functional derivative method to compute the covariance of two smooth linear statistics for the one-dimensional Riesz gas with exponent s ∈ (-1,1), s ≠ 0, in the infinite-density limit with particle support restricted to [-1,1]. The covariance is expressed as a sum over Fourier components of the linear statistics in the Gegenbauer polynomial basis {C_n^{(s/2)}(x)}, generalizing the cosine expansion known for the s=0 logarithmic case. For power-sum linear statistics the general formula reduces to a product of gamma functions that is compared to recent exact results in the literature.

Significance. If the central derivation holds, the result supplies an explicit, basis-diagonalized expression for fluctuations in Riesz gases that interpolates between the logarithmic and power-law regimes. The match of the power-sum reduction to independent exact results provides a concrete consistency check, and the use of the equilibrium measure's weight to ensure orthogonality of the Gegenbauer polynomials is a natural structural feature. This could serve as a reference formula for correlation studies in one-dimensional long-range interacting systems.

minor comments (3)

Abstract: the statement that the power-sum reduction 'can be reduced to a product of gamma function form' should be accompanied by the explicit formula (or a reference to the equation number where it appears) so that the comparison with literature results is immediately verifiable.
The manuscript should include a short outline of the key steps in Beenakker's functional derivative technique as applied here, even if the original reference is cited, to make the passage from the equilibrium measure to the diagonal covariance formula self-contained.
Notation: the precise definition of the 'Fourier components' of a linear statistic with respect to the Gegenbauer basis should be stated explicitly (including any normalization constants) before the covariance sum is written.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation for minor revision. We appreciate the recognition that the Gegenbauer expansion provides a useful reference formula interpolating between regimes.

read point-by-point responses

Referee: The manuscript applies Beenakker's functional derivative method to compute the covariance of two smooth linear statistics for the one-dimensional Riesz gas with exponent s ∈ (-1,1), s ≠ 0, in the infinite-density limit with particle support restricted to [-1,1]. The covariance is expressed as a sum over Fourier components of the linear statistics in the Gegenbauer polynomial basis {C_n^{(s/2)}(x)}, generalizing the cosine expansion known for the s=0 logarithmic case. For power-sum linear statistics the general formula reduces to a product of gamma functions that is compared to recent exact results in the literature.

Authors: We thank the referee for this accurate summary of the main results. The use of the equilibrium measure to ensure orthogonality of the Gegenbauer polynomials is indeed a key structural feature of the derivation, and the reduction to gamma-function products for power sums serves as the consistency check with independent exact results. revision: no

Circularity Check

0 steps flagged

No significant circularity; derivation grounded in external method and independent checks

full rationale

The paper applies Beenakker's external functional derivative technique to derive the covariance of linear statistics for the Riesz gas in the infinite-density limit on [-1,1]. The resulting expression is expanded in the Gegenbauer basis C_n^{(s/2)}(x), which is the natural orthogonal basis for the equilibrium weight (1-x^2)^{(s-1)/2} induced by the Riesz kernel; this choice generalizes the s=0 cosine case without self-definition. The algebraic reduction of the power-sum case to gamma-function products is a direct consequence of the formula and is validated by matching independent exact results from the literature, providing external grounding. No steps reduce by construction to fitted inputs, self-citations, or tautological definitions; the central claim is a computed explicit formula rather than a re-expression of its assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the applicability of Beenakker's functional derivative technique to the Riesz potential and on standard properties of Gegenbauer polynomials as an orthogonal basis.

axioms (2)

domain assumption Beenakker's functional derivative method yields the covariance for the Riesz pair potential in the infinite-density limit
Invoked to obtain the stated sum over Gegenbauer components
standard math Gegenbauer polynomials C_n^{(s/2)}(x) form a complete orthogonal basis on [-1,1] for the relevant weight
Used to expand the linear statistics

pith-pipeline@v0.9.0 · 5467 in / 1444 out tokens · 73171 ms · 2026-05-07T14:54:09.864364+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

41 extracted references · 3 canonical work pages

[1]

Abdou,Spectral relationships for integral operators in contact problem of impressing stamps, Appl

M.A. Abdou,Spectral relationships for integral operators in contact problem of impressing stamps, Appl. Math. Computat.118(2001), 95-111

2001
[2]

Agarwal, A

S. Agarwal, A. Dhar, M. Kulkarni, A. Kundu, S. N. Majumdar, D. Mukamel, and G. Schehr, Harmonically confined particles with long-range repulsive interactions, Phys. Rev. Lett.123 (2019), 100603

2019
[3]

Andrews, R.J

G.E. Andrews, R.J. Baxter and P.J. Forrester,Eight-vertex SOS model and generalized Rogers-Ramanujan-type identitiesJ. Stat. Phys.35(1984), 193–266

1984
[4]

Atai,A kernel function approach to exact solutions of Calogero-Moser-Sutherland type models, PhD thesis, KTH, 2016

F. Atai,A kernel function approach to exact solutions of Calogero-Moser-Sutherland type models, PhD thesis, KTH, 2016

2016
[5]

Baxter,Statistical mechanics of a one-dimensional Coulomb system with a uniform charge background, Math

R.J. Baxter,Statistical mechanics of a one-dimensional Coulomb system with a uniform charge background, Math. Proc. Cambridge Philos. Soc.,59(1963), 779–787

1963
[6]

Baxter,Dimers on a rectangular lattice, J

R.J. Baxter,Dimers on a rectangular lattice, J. Math. Phys.9(1968), 650–654

1968
[7]

Baxter,Exactly solved models in statistical mechanics, Academic Press, Lodon, 1982

R.J. Baxter,Exactly solved models in statistical mechanics, Academic Press, Lodon, 1982

1982
[8]

Baxter,Partition function of the three-dimensional Zamolodchikov model, Phys

R.J. Baxter,Partition function of the three-dimensional Zamolodchikov model, Phys. Rev. Lett.53(1984), 1795–1798

1984
[9]

Baxter,Derivation of the order parameter of the chiral Potts model, Phys

R.J. Baxter,Derivation of the order parameter of the chiral Potts model, Phys. Rev. Lett. 94(2005), 130602

2005
[10]

Baxter,The order parameter of the chiral Potts model, J

R.J. Baxter,The order parameter of the chiral Potts model, J. Stat. Phys.120, 1–36
[11]

Baxter,The challenge of the chiral Potts model, Journal of Physics: Conference Series 42(2006), 11–24

R.J. Baxter,The challenge of the chiral Potts model, Journal of Physics: Conference Series 42(2006), 11–24. 12

2006
[12]

Baxter,Some academic and personal reminiscences of Rodney James Baxter, J

R.J. Baxter,Some academic and personal reminiscences of Rodney James Baxter, J. Phys. A48(2015), 254001

2015
[13]

Baxter and P.J

R.J. Baxter and P.J. Forrester,Is the Zamolodchikov model critical?, J. Phys. A18, 1483— 1497
[14]

Beenakker,Random-matrix theory of mesoscopic fluctuations in conductors and superconductors, Phys

C.W.J. Beenakker,Random-matrix theory of mesoscopic fluctuations in conductors and superconductors, Phys. Rev. B47(1993), 15763–15775

1993
[15]

Beenakker,Pair correlation function of the one-dimensional Riesz gas, Phys

C.W.J. Beenakker,Pair correlation function of the one-dimensional Riesz gas, Phys. Rev. Research5, 013152 (2023)

2023
[16]

Boursier,Optimal local laws and CLT for the circular Riesz gas, arXiv:2112.05881

J. Boursier,Optimal local laws and CLT for the circular Riesz gas, arXiv:2112.05881

work page arXiv
[17]

Byun and P.J

S.-S. Byun and P.J. Forrester,Electrostatic computations for statistical mechanics and random matrix applications, arXiv:2510.14334

work page arXiv
[18]

Byun, P.J

S.-S. Byun, P.J. Forrester, S.N. Majumdar and G. Schehr,Equilibrium measures for higher dimensional rotationally symmetric Riesz gases, arXiv:2602.03047

work page arXiv
[19]

Dandekar, P.L

R. Dandekar, P.L. Krapivsky, and K. Mallick,Dynamical fluctuations in the Riesz gas, Physical Review E,107(2023), 044129

2023
[20]

Desrosiers and D.-Z

P. Desrosiers and D.-Z. Liu, Asymptotics for products of characteristic polynomials in classicalβ-ensembles, Constr. Approx.39(2014), 273

2014
[21]

Durand, P.M

L. Durand, P.M. Fishbane, and L.M. Simmons Jr.,Expansion formulas and addition theo- rems for Gegenbauer functions, J. Math. Physics,17(1976), 1933-1948

1976
[22]

Fahmy, M.A

M.H. Fahmy, M.A. Abdou, M.A. Darwish,Integral equations and potential-theoretic type integrals of orthogonal polynomials, J. Comp. Appl. Math.,106(1999), 245-254

1999
[23]

Flack, S

A. Flack, S. N. Majumdar, and G. Schehr,An exact formula for the variance of linear statistics in the one-dimensional jellium model, J. Phys. A56, 105002 (2023)

2023
[24]

Forrester,Log-gases and random matrices, Princeton University Press, Princeton, NJ, 2010

P.J. Forrester,Log-gases and random matrices, Princeton University Press, Princeton, NJ, 2010

2010
[25]

Forrester,A review on exact results for fluctuation formulas in random matri theory, Probability Surveys20(2023), 170–225

P.J. Forrester,A review on exact results for fluctuation formulas in random matri theory, Probability Surveys20(2023), 170–225

2023
[26]

Forrester, R.J

P.J. Forrester, R.J. Baxter,Further exact solutions of the eight-vertex SOS model and generalizations of the Rogers-Ramanujan identities. J Stat Phys38(1985), 435–472

1985
[27]

Forrester, N.E

P.J. Forrester, N.E. Frankel, and T.M. Garoni,Random matrix averages and the impene- trable Bose gas in Dirichlet and Neumann boundary conditions, J. Math. Phys,44(2003), 4157

2003
[28]

Gaudin,Sur la loi limite de l’espacement des valeurs propres d’une matrice al´ eatoire, Nucl

M. Gaudin,Sur la loi limite de l’espacement des valeurs propres d’une matrice al´ eatoire, Nucl. Phys.25(1961), 447-458

1961
[29]

Gradshteyn and I.M

I.S. Gradshteyn and I.M. Ryzhik,Table of Integrals, Series, and Products, 7th edition, Academic Press, New Yorkm 2007,

2007
[30]

D. P. Hardin, T. Lebl´ e, E. B. Saff, and S. Serfaty,Large deviation principles for hypersin- gular Riesz gases, Constr. Approx.48(2018), 61. 13

2018
[31]

Kethepalli, M

J. Kethepalli, M. Kulkarni, A. Kundu, S. N. Majumdar, D. Mukamel, and G. Schehr, Harmonically confined long-ranged interacting gas in the presence of a hard wall, J. Stat. Mech.2021(2021), 103209

2021
[32]

Kethepalli, M

J. Kethepalli, M. Kulkarni, A. Kundu, S.N. Majumdar, D. Mukamel and G. Schehr,Edge fluctuations and third-order phase transition in harmonically confined long-range systems, J. Stat. Mech. Theory Exp.2022(2022), 033203

2022
[33]

Le Doussal and G

P. Le Doussal and G. Schehr, Cumulants and large deviations for the linear statistics of the one-dimensional trapped Riesz gas, J. Stat. Phys.192(2025), 47

2025
[34]

Lebl´ e and S

T. Lebl´ e and S. Serfaty,Large deviation principle for empirical fields of Log and Riesz gases, Invent. Math.210(2017), 645

2017
[35]

Lenard,Exact statistical mechanics of a one-dimensional system with Coulomb forces, J

A. Lenard,Exact statistical mechanics of a one-dimensional system with Coulomb forces, J. Math. Phys.,2(1961), 682–695

1961
[36]

Lewin,Coulomb and Riesz gases: the known and the unknown, J

M. Lewin,Coulomb and Riesz gases: the known and the unknown, J. Math. Phys.63 (2022), 061101

2022
[37]

Mkhitarian and M.A

S.M. Mkhitarian and M.A. Abdou,On various method for the solution of the Carleman integral equation, Dakl. Acad. Nauk Arm. SSR,89(1990), 125-129

1990
[38]

Porter and D.S.G

D. Porter and D.S.G. Stirling,Integral equations: a practical treatment, from spectral theory to applications, Cambridge University Press, Cambridge, 1990

1990
[39]

Rainville,Special functions, The Macmillan Company, New York, 1960

E.D. Rainville,Special functions, The Macmillan Company, New York, 1960

1960
[40]

Shen and P.J

B.-J. Shen and P.J. Forrester,Moments of characteristic polynomials for classicalβen- sembles, J. Math. Phys.66(2025), 113301

2025
[41]

Touzo, P

L. Touzo, P. Le Doussal, G. Schehr,Spatio-temporal fluctuations in the passive and active Riesz gas on the circle, J. Stat. Phys.192(2025), 79. 14

2025