Gessel-Type Expansion for the Circular $\beta$-Ensemble and Central Limit Theorem for the Sine-$\beta$ Process for $\beta\le 2$

Sergei M. Gorbunov

REVIEW 2 major objections 1 minor 50 references

Gessel-Type Expansion for the Circular $\beta$-Ensemble and Central Limit Theorem for the Sine-$\beta$ Process for $\beta\le 2$

T0 review · 2 major / 1 minor · reviewed 2026-05-18 · grok-4.3

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Pith's one-line read 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.

desk verdict The paper gives a Gessel-type expansion in Jack polynomials for circular beta-ensemble expectations that extends Szego limits and sine-beta CLTs to beta <=2 in the full H^{1/2} class, but the uniformity of the series control is the part to check. read the letter →

arxiv 2510.15172 v2 submitted 2025-10-16 math.PR math-phmath.COmath.MP

Sergei M. Gorbunov This is my paper · ORCID

classification math.PRmath-phmath.COmath.MP

keywords circularbetaensembleJackpolynomialsSzegotheoremcentrallimitsine-betaprocessmultiplicativefunctionalsrandompointprocesses

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

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.

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.

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.

Watch

Extended reading notes

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.

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.

Editorial extensions

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.

Reading between the lines

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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

read the letter

The main point is that this work supplies a Gessel-style series expansion in Jack polynomials for expectations of multiplicative functionals under the circular beta-ensemble measure. From there it derives a Szego-type asymptotic for every test function in H^{1/2}(T) when beta is at most 2, with an explicit rate for the smoother H^1 case, and shows the same estimates survive the scaling limit to produce a Soshnikov-style central limit theorem for the sine-beta process on the full H^{1/2}(R) class. For beta=2 the construction reduces to the familiar Schur/Toeplitz setting, so the new ground is the lower-beta range. The expansion itself looks like the concrete technical step that makes the rest possible, and carrying the remainder through the scaling limit is a clean move if the bounds are right. The paper therefore gives a usable combinatorial handle on a range of beta that has been awkward for these limit theorems. The soft spot is exactly the one the stress-test note flags: whether the series in Jack polynomials stays absolutely convergent or summable with an explicit remainder that works uniformly across the whole H^{1/2} class when beta drops below 2. The hook-length factors and Jack norms change their growth, and if the control only holds on a dense subclass or needs extra decay on the test function, the headline statements for the full Sobolev class would need to be weakened. The abstract gives no derivation steps or sample bounds, so the referee would have to look at those estimates directly. This is aimed at people who already work on beta-ensembles, point-process limits, and Szego-type results. A reader who wants a new tool for handling multiplicative functionals in the low-beta regime would get something concrete out of it. The claims are specific enough and the extension is potentially useful, so the paper deserves a serious referee to check the expansion and the remainder estimates. I would send it to peer review.

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)

[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.
[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)

[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

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
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 · score 0.0 of 10

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.

Assumptions & free parameters 0 free parameters · 1 assumptions · 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.

assumptions (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.

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Cite this review

Pith. "Pith review of Gessel-Type Expansion for the Circular $\beta$-Ensemble and Central Limit Theorem for the Sine-$\beta$ Process for $\beta\le 2$." pith.science (2026). https://pith.science/paper/2510.15172

@misc{pith2026251015172,
  author       = {Pith},
  title        = {Pith review of: Gessel-Type Expansion for the Circular $\beta$-Ensemble and Central Limit Theorem for the Sine-$\beta$ Process for $\beta\le 2$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2510.15172}},
  note         = {Machine review of arXiv:2510.15172}
}

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.

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed May 18, 2026 · model on record in the stance chip above.

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