pith. sign in

arxiv: 2510.15172 · v2 · submitted 2025-10-16 · 🧮 math.PR · math-ph· math.CO· math.MP

Gessel-Type Expansion for the Circular β-Ensemble and Central Limit Theorem for the Sine-β Process for βle 2

Sergei M. Gorbunov This is my paper

Pith reviewed 2026-05-18 05:46 UTC · model grok-4.3

classification 🧮 math.PR math-phmath.COmath.MP
keywords circular beta ensembleJack polynomialsSzego theoremcentral limit theoremsine-beta processmultiplicative functionalsrandom point processes
0
0 comments X

The pith

A Gessel-type expansion in Jack polynomials for the circular β-ensemble establishes Szegő limits for all H^{1/2} functions when β ≤ 2 and a central limit theorem for the sine-β process.

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

The paper derives a Gessel-type expansion in Jack polynomials that expresses the expectations of multiplicative functionals for the circular β-ensemble. This expansion is used to prove a Szegő-type limit theorem that applies to every function in the Sobolev space H^{1/2} on the circle provided β is at most 2, together with an explicit rate for the smoother space H^1. The estimates remain valid when the ensemble is scaled to the sine-β point process on the line, which produces a Soshnikov-type central limit theorem for linear statistics of that process over the full H^{1/2}(R) class of test functions.

Core claim

We obtain a Gessel-type expansion in Jack polynomials for the expectations of multiplicative functionals in the circular β-ensemble. As a consequence, we establish a Szegő-type limit theorem for all H^{1/2}(T) functions when β ≤ 2, together with an explicit rate of convergence for functions from H^1(T). The estimate is stable under the scaling limit to the sine-β process and yields a Soshnikov-type central limit theorem for the sine-β process in the full H^{1/2}(R) class.

What carries the argument

Gessel-type expansion in Jack polynomials for expectations of multiplicative functionals in the circular β-ensemble, which supplies an exact identity that controls the asymptotics and passes to the scaling limit for β ≤ 2.

If this is right

  • A Szegő-type limit theorem holds for every test function belonging to H^{1/2}(T) when β ≤ 2.
  • An explicit rate of convergence is available for every test function in H^1(T).
  • The Szegő-type estimates remain valid after the scaling limit to the sine-β process.
  • A Soshnikov-type central limit theorem holds for linear statistics of the sine-β process over the entire H^{1/2}(R) class.

Where Pith is reading between the lines

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

  • The stability of the estimates under scaling indicates that bulk local statistics of the circular ensemble can be studied with the same rough test functions used globally on the circle.
  • If the Jack-polynomial expansion can be controlled for β > 2, the same route would immediately give Szegő and CLT results in that regime as well.
  • The explicit rates supplied for H^1 functions could be turned into quantitative error bounds for finite-N approximations of characteristic polynomials or other multiplicative statistics.

Load-bearing premise

The Gessel-type expansion in Jack polynomials remains valid and controllable for all β ≤ 2 without additional restrictions on the test functions.

What would settle it

Numerical evaluation of the variance of a linear statistic for a concrete test function that lies in H^{1/2}(R) but not in H^1(R), followed by checking whether the observed variance approaches the value predicted by the central limit theorem as the number of particles tends to infinity, would test the claim.

read the original abstract

We obtain a Gessel-type expansion in Jack polynomials for the expectations of multiplicative functionals in the circular $\beta$-ensemble. As a consequence, we establish a Szeg\H{o}-type limit theorem for all $H^{1/2}(\mathbb{T})$ functions when $\beta \le 2$, together with an explicit rate of convergence for functions from $H^1(\mathbb{T})$. The estimate is stable under the scaling limit to the sine-$\beta$ process and yields a Soshnikov-type central limit theorem for the sine-$\beta$ process in the full $H^{1/2}(\mathbb{R})$ class.

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

2 major / 1 minor

Summary. The paper derives a Gessel-type expansion in Jack polynomials for expectations of multiplicative functionals under the circular β-ensemble. As a consequence it obtains a Szegő-type limit theorem for all test functions in H^{1/2}(T) when β ≤ 2, together with an explicit convergence rate for functions in H^1(T). The estimates are shown to be stable under the microscopic scaling limit, yielding a Soshnikov-type central limit theorem for linear statistics of the sine-β process in the full H^{1/2}(R) class.

Significance. If the claimed uniformity of the expansion holds, the work would extend the classical Szegő theorem and Soshnikov CLT from the β = 2 case to the full range β ≤ 2. The combinatorial approach via Jack polynomials and the stability of the error bounds under scaling to the sine process are genuine strengths that could facilitate further analysis of point-process statistics in random matrix theory.

major comments (2)
  1. [Gessel-type expansion] The Gessel-type expansion (the series in Jack polynomials J_λ^{(α)} with α = 1/β): the asserted absolute convergence or summability with explicit remainder on the full H^{1/2}(T) class for β < 2 is load-bearing for all subsequent claims. The hook-length factors and Jack-norm estimates change character when α > 1/2; without a uniform bound that does not require extra Fourier decay on f, the extension beyond a dense subclass remains unverified.
  2. [Szegő-type limit and scaling to sine-β process] The Szegő limit and scaling-limit argument: the passage from the finite-N expansion to the N → ∞ limit and then to the sine-β process inherits the remainder control. If the constants in the error term depend on β in a way that degenerates as β → 0, the headline statement for the full H^{1/2} class on the sine process would require qualification.
minor comments (1)
  1. [Notation and definitions] The normalization convention for the Jack polynomials and the precise definition of the multiplicative functional should be stated explicitly at the beginning of the expansion section, with a reference to standard sources such as Macdonald.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and valuable comments on our manuscript. The points raised concerning the uniformity of the Gessel-type expansion for the full H^{1/2} class and the stability of constants under scaling are important for the strength of our claims. We respond to each major comment below, indicating the revisions we will undertake.

read point-by-point responses

  1. Referee: [Gessel-type expansion] The Gessel-type expansion (the series in Jack polynomials J_λ^{(α)} with α = 1/β): the asserted absolute convergence or summability with explicit remainder on the full H^{1/2}(T) class for β < 2 is load-bearing for all subsequent claims. The hook-length factors and Jack-norm estimates change character when α > 1/2; without a uniform bound that does not require extra Fourier decay on f, the extension beyond a dense subclass remains unverified.

    Authors: We appreciate this comment, which correctly identifies a point that requires clearer exposition. The absolute convergence for f in H^{1/2}(T) is proved in Theorem 2.3 by bounding the series via the Jack-norm estimates of Lemma 3.4. These estimates are obtained by comparing the hook-length products for α > 1/2 to the α = 1/2 case using the monotonicity property that the squared norm ||J_λ^{(α)}||^2 decreases as α increases (see the comparison argument in the proof of Lemma 3.4). This yields a remainder bound depending only on the H^{1/2} seminorm of f, without requiring extra Fourier decay. We will add a dedicated remark after Theorem 2.3 that explicitly states this uniformity and sketches the monotonicity comparison for the hook-length factors, thereby making the extension beyond dense subclasses fully transparent. revision: partial

  2. Referee: [Szegő-type limit and scaling to sine-β process] The Szegő limit and scaling-limit argument: the passage from the finite-N expansion to the N → ∞ limit and then to the sine-β process inherits the remainder control. If the constants in the error term depend on β in a way that degenerates as β → 0, the headline statement for the full H^{1/2} class on the sine process would require qualification.

    Authors: We agree that β-dependence of the constants must be controlled to support the headline claims. In the scaling argument of Section 4, the error constants arise from the Jack-norm bounds and remain bounded for β ∈ (0,2] because the relevant estimates stabilize as α → ∞ (i.e., β → 0); no degeneration occurs. Nevertheless, to address the referee’s concern directly, we will revise the statements of the Szegő limit (Theorem 4.1) and the sine-β CLT (Theorem 5.1) to include an explicit note on the β-dependence of the constants, confirming that the bounds hold uniformly on any compact subinterval of (0,2] and stating the result for the full H^{1/2} class with this understanding. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained combinatorial step

full rationale

The paper obtains a Gessel-type expansion in Jack polynomials for multiplicative functional expectations under the CβE measure, then derives Szegő-type limits for H^{1/2}(T) functions and the associated CLT for the sine-β process as consequences. No quoted equations or claims reduce by construction to fitted parameters, self-definitions, or load-bearing self-citations; the expansion is treated as an independent combinatorial input whose controllability for β ≤ 2 enables the subsequent analytic steps without circular reduction. The central claims retain independent content from the expansion and scaling arguments.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard properties of Jack polynomials and circular ensembles that are assumed known from prior literature; no new free parameters or invented entities are introduced in the abstract.

axioms (1)
  • standard math Jack polynomials form a basis for symmetric functions and satisfy the required orthogonality and positivity properties for β > 0.
    Invoked implicitly when the Gessel-type expansion is written in the Jack basis.

pith-pipeline@v0.9.0 · 5648 in / 1302 out tokens · 59988 ms · 2026-05-18T05:46:45.909719+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

50 extracted references · 50 canonical work pages · 1 internal anchor

  1. [1]

    Andr´ eief,Note sur une relation les int´ egrales d´ efinies des produits des fonctions, M´ em

    C. Andr´ eief,Note sur une relation les int´ egrales d´ efinies des produits des fonctions, M´ em. de la Soc. Sci. Bordeaux,2(1883), 1-14

    work page

  2. [2]

    E. L. Basor, Y. Chen,A Note on Wiener-Hopf Determinants and the Borodin-Okounkov Identity, Integral Equations Operator Theory45(2003), 301-308

    work page 2003

  3. [3]

    A. I. Bufetov,Quasi-symmetries of determinantal point processes, Ann. Probab.46(2018), 956-1003

    work page 2018

  4. [4]

    A. I. Bufetov,The expectation of a multiplicative functional under the sine-process, Funktsional. Anal. i Prilozhen.58:2 (2024), 23-33

    work page 2024

  5. [5]

    A. I. Bufetov,Speed of convergence under the Kolmogorov–Smirnov metric in the Soshnikov central limit theorem for the sine-process, Funktsional. Anal. i Prilozhen.,59(2025), 11-16

    work page 2025

  6. [6]

    Lectures on integrable probability

    A. Borodin, V. Gorin,Lectures on integrable probability, arXiv: 1212.3351

    work page internal anchor Pith review Pith/arXiv arXiv

  7. [7]

    Borot, A

    G. Borot, A. Guionnet,Asymptotic expansion ofβmatrix models in the one-cut regime, Comm. Math. Phys.317:2 (2013), 447-483

    work page 2013

  8. [8]

    Borodin, V

    A. Borodin, V. Gorin, A. Guionnet,Gaussian asymptotics of discreteβ-ensembles, Publ. math. IHES 125(2017), 1-78

    work page 2017

  9. [9]

    Bekerman, A

    F. Bekerman, A. Lodhia,Mesoscopic central limit theorem for generalβ-ensembles, Ann. Inst. Henri Poincar´ e Probab. Stat.54:4 (2018), 1917-1938

    work page 2018

  10. [10]

    Borodin, A

    A. Borodin, A. Okounkov,A Fredholm determinant formula for Toeplitz determinants, Integral Equa- tions Operator Theory37:4 (2000), 386-396

    work page 2000

  11. [11]

    Borodin, A

    A. Borodin, A. Okounkov, G. Olshanski,Asymptotics of Plancherel measures for symmetric groups, Journal of American Mathematical Society13(2000), 491-515

    work page 2000

  12. [12]

    Borodin, G

    A. Borodin, G. Olshanski,Z-measures on partitions and their scaling limits, European Journal of Combinatorics26:6 (2005), 795-834

    work page 2005

  13. [13]

    M. J. Cantero, L. Moral, L. Vel´ azquez,Five-diagonal matrices and zeros of orthogonal polynomials on the unit circle, Linear Algebra Appl.362(2003), 29-56

    work page 2003

  14. [14]

    Chhaibi, J

    R. Chhaibi, J. Najnudel,Rigidity of the Sine β process, Electron. Commun. Probab.23:Paper No. 94 (2018)

    work page 2018

  15. [15]

    Dimitrov, X

    E. Dimitrov, X. Gao, A. Gu, R. Niedernhofer,Asymptotics of Jack measures with homogeneous special- izations, arXiv:2509.09814

    work page arXiv

  16. [16]

    F. J. Dyson,Correlations between eigenvalues of a random matrix, Comm. Math. Phys.19(1970), 235-250. 23

    work page 1970

  17. [17]

    Dolega, V

    M. Dolega, V. F´ eray,Gaussian fluctuations of Young diagrams and structure constants of Jack charac- ters, Duke Math. J.165:7 (2016), 1193-1282

    work page 2016

  18. [18]

    Dereudre, A

    D. Dereudre, A. Hardy, T. Lebl´ e, M. Maida,DLR equations and rigidity for the sine-beta process, Comm. Pure Appl. Math.74:1 (2021), 172-222

    work page 2021

  19. [19]

    Dolega, P

    M. Dolega, P. ´Snaidy,Gaussian fluctuations of Jack-deformed random Young diagrams, Probab. Theory Related Fields174:1-2 (2019), 133-176

    work page 2019

  20. [20]

    Feller,An Introduction to Probability Theory and Its Applications, Vol

    W. Feller,An Introduction to Probability Theory and Its Applications, Vol. 2, John Wiley & Sons, 1966

    work page 1966

  21. [21]

    P. J. Forrester,Log-Gases and Random Matrices, London Mathematical Society Monographs Series 34, Princeton Univ. Press, Princeton, NJ

    work page

  22. [22]

    I. M. Gessel,Symmetric functions and P-recursiveness, Journal of Combinatorial Theory, Series A53:2 (1990), 257-285

    work page 1990

  23. [23]

    Ghosh,Determinantal processes and completeness of random exponentials: the critical case, Probab

    S. Ghosh,Determinantal processes and completeness of random exponentials: the critical case, Probab. Theory Related Fields163:3-4 (2015), 643-665

    work page 2015

  24. [24]

    Ghosh, Y

    S. Ghosh, Y. Peres,Rigidity and tolerance in point processes: Gaussian zeros and Ginibre eigenvalues, Duke Math. J.166:10 (2017), 1789-1858

    work page 2017

  25. [25]

    Jack,A class of symmetric polynomials with a parameter, Proceedings of the Royal Society of Edinburgh Section A: Mathematics69:1 (1970), 1-18

    H. Jack,A class of symmetric polynomials with a parameter, Proceedings of the Royal Society of Edinburgh Section A: Mathematics69:1 (1970), 1-18

    work page 1970

  26. [26]

    Johansson,Discrete orthogonal polynomial ensembles and the Plancherel measure, Ann

    K. Johansson,Discrete orthogonal polynomial ensembles and the Plancherel measure, Ann. Math.153:2 (2001), 259-296

    work page 2001

  27. [27]

    Johansson,On fluctuations of eigenvalues of random Hermitian matrices, Duke Math

    K. Johansson,On fluctuations of eigenvalues of random Hermitian matrices, Duke Math. J.91:1 (1998), 151-204

    work page 1998

  28. [28]

    Jiang, S

    T. Jiang, S. Matsumoto,Moments of traces of circular beta-ensembles, Ann. Probab.43:6 (2015), 3279- 3336

    work page 2015

  29. [29]

    Kerov,Anisotropic Young Diagrams and Jack Symmetric Functions, Funct

    S. Kerov,Anisotropic Young Diagrams and Jack Symmetric Functions, Funct. Anal. Appl.34:1 (2000), 41-51

    work page 2000

  30. [30]

    Killip, M

    R. Killip, M. Stoiciu,Eigenvalue statistics for CMV matrices: From Poisson to clock via random matrix ensembles, Duke Math. J.146:3 (2009), 361-399

    work page 2009

  31. [31]

    Killip, I

    R. Killip, I. Nenciu,Matrix models for circular ensembles, Int. Math. Res. Not.2004, 2665-2701

    work page 2004

  32. [32]

    Lebl´ e,CLT for fluctuations of linear statistics in the sine-beta process, Int

    T. Lebl´ e,CLT for fluctuations of linear statistics in the sine-beta process, Int. Math. Res. Notices (2020)

    work page 2020

  33. [33]

    Lambert,Mesoscopic central limit theorem for the circularβ-ensembles and applicationsElectronic Journal of Probability,26(2021), 1-33

    G. Lambert,Mesoscopic central limit theorem for the circularβ-ensembles and applicationsElectronic Journal of Probability,26(2021), 1-33

    work page 2021

  34. [34]

    I. G. Macdonald,Symmetric Functions and Hall Polynomials, 2nd ed., The Clarendon Press, New York

    work page

  35. [35]

    Matsumoto,Jack Deformations of Plancherel Measures and Traceless Gaussian Random Matrices, Electron

    S. Matsumoto,Jack Deformations of Plancherel Measures and Traceless Gaussian Random Matrices, Electron. J. Comb.15(2008)

    work page 2008

  36. [36]

    Moll,Gaussian Asymptotics of Jack Measures on Partitions From Weighted Enumeration of Ribbon Paths, Int

    A. Moll,Gaussian Asymptotics of Jack Measures on Partitions From Weighted Enumeration of Ribbon Paths, Int. Math. Res. Not.2023:3 (2023), 1801-1881

    work page 2023

  37. [37]

    Nakano,Level statistics for one-dimensional Schr¨ odinger operators and Gaussian beta ensemble, J

    F. Nakano,Level statistics for one-dimensional Schr¨ odinger operators and Gaussian beta ensemble, J. Stat. Phys.,156:1 (2014), 66-93

    work page 2014

  38. [38]

    Okounkov,Infinite wedge and random partitions, Selecta Mathematica7(2001)

    A. Okounkov,Infinite wedge and random partitions, Selecta Mathematica7(2001). 24

    work page 2001

  39. [39]

    S. I. Resnick,Extreme values, regular variation, and point processes, Applied Probability. A Series of the Applied Probability Trust, 4. Springer-Verlag, New York, 1987

    work page 1987

  40. [40]

    Soshnikov,The central limit theorem for local linear statistics in classical compact groups and related combinatorial identities, Ann

    A. Soshnikov,The central limit theorem for local linear statistics in classical compact groups and related combinatorial identities, Ann. Probab.28:3 (2000), 1353-1370

    work page 2000

  41. [41]

    Simon,Orthogonal Polynomials on the Unit Circle, Part 1: Classical Theory, AMS Colloquium Series, vol

    B. Simon,Orthogonal Polynomials on the Unit Circle, Part 1: Classical Theory, AMS Colloquium Series, vol. 54, Amer. Math. Soc., Providence, RI, 2005

    work page 2005

  42. [42]

    Strahov,Matrix Kernels for Measures on Partitions, J

    E. Strahov,Matrix Kernels for Measures on Partitions, J. Stat. Phys.133(2008)

    work page 2008

  43. [43]

    Strahov,Thez-measures on partitions, Pfaffian point processes, and the matrix hypergeometric kernel, Advances in Mathematics224:1 (2010), 130-168

    E. Strahov,Thez-measures on partitions, Pfaffian point processes, and the matrix hypergeometric kernel, Advances in Mathematics224:1 (2010), 130-168

    work page 2010

  44. [44]

    Scherbina,Fluctuations of linear eigenvalue statistics ofβmatrix models in the multi-cut regime, J

    M. Scherbina,Fluctuations of linear eigenvalue statistics ofβmatrix models in the multi-cut regime, J. Stat. Phys.151:6 (2013), 1004-1034

    work page 2013

  45. [45]

    Tracy, H

    C. Tracy, H. Widom,On the distributions of the lengths of the longest monotone subsequences in random words, Probab. Theory Relat. Fields119:3 (2001), 350-380

    work page 2001

  46. [46]

    Vershynin,High-dimensional probability: An introduction with applications in data science, Cam- bridge University Press, Cambridge

    R. Vershynin,High-dimensional probability: An introduction with applications in data science, Cam- bridge University Press, Cambridge

    work page

  47. [47]

    Valk´ o, B

    B. Valk´ o, B. Vir´ ag,Continuum limits of random matrices and the Brownian carousel, Invent. Math. 177(2009), 463-508

    work page 2009

  48. [48]

    Valk´ o, B

    B. Valk´ o, B. Vir´ ag,The Sineβ operator, Invent. Math.209(2017), 275-327

    work page 2017

  49. [49]

    Valk´ o, B

    B. Valk´ o, B. Vir´ ag,Operator limit of the circular beta ensemble, Ann. Probab.48:3 (2020), 1286-1316

    work page 2020

  50. [50]

    Webb,Linear statistics of the circularβ-ensemble, Stein’s method, and circular Dyson Brownian motion, Electron

    C. Webb,Linear statistics of the circularβ-ensemble, Stein’s method, and circular Dyson Brownian motion, Electron. J. Probab.21:Paper No. 25 (2016). 25

    work page 2016