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arxiv: 1906.12130 · v1 · pith:XGW3KNGNnew · submitted 2019-06-28 · 🧮 math-ph · math.MP

Higher order large gap asymptotics at the hard edge for Muttalib--Borodin ensembles

Christophe Charlier , Jonatan Lenells , Julian Mauersberger This is my paper

Pith reviewed 2026-05-25 13:40 UTC · model grok-4.3

classification 🧮 math-ph math.MP
keywords Muttalib-Borodin ensembleshard edgelarge gap asymptoticsWright generalized Bessel functionsBarnes G-functiongap probabilitiespoint processes
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The pith

The large gap asymptotics for Muttalib-Borodin ensembles at the hard edge extend to all orders in s, with explicit expressions for constants c and C obtained from a differential identity in θ.

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

This paper derives expressions for the subleading constants in the asymptotic expansion of large gap probabilities for the hard-edge limiting process of Muttalib-Borodin ensembles. The expansion takes the form of an exponential with a power-law prefactor, and the authors evaluate the logarithmic and constant terms using differentiation with respect to the ensemble parameter θ. For rational values of θ they express the overall constant factor using the Barnes G-function. They further prove that the expansion holds with corrections of all orders in the gap size s.

Core claim

The probability of a gap on the interval [0,s] in the limiting hard-edge point process admits the asymptotic form C exp(−a s^{2ρ} + b s^ρ + c ln s) times (1 + higher order terms in s) as s tends to infinity, where the constants a, b, ρ were known and c, C are now determined by applying a differential identity with respect to θ to the gap probability; when θ is a positive rational number, C is given explicitly in terms of the Barnes G-function.

What carries the argument

Differential identity in the parameter θ applied after the first- and second-order asymptotics have been established via Riemann-Hilbert analysis of the kernel built from Wright's generalized Bessel functions.

Load-bearing premise

The differential identity in θ that is used to evaluate c and C remains valid for the limiting kernel built from Wright's generalized Bessel functions.

What would settle it

Direct numerical evaluation of the gap probability for large s with a specific rational θ, such as θ=1, compared against the explicit formula involving Barnes' G-function.

Figures

Figures reproduced from arXiv: 1906.12130 by Christophe Charlier, Jonatan Lenells, Julian Mauersberger.

Figure 1
Figure 1. Figure 1: The contours γ and ˜γ for α = 1.6 and θ = 1.2. The dots are the zeros and poles of F. double contour integral expression (from [12]) will be important: K(x, y) = 1 4π 2 Z γ du Z γ˜ dvF(u) F(v) x −uy v−1 u − v , x, y > 0, (1.3) where the function F is given by F(z) = Γ [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The points b1 and b2 lie in the upper half-plane for 0 < θ < 1. The contour Σ5 consists of the two line segments [b1, 0] and [0, b2]. RH problem for Y (a) Y : C \ (γ ∪ γ˜) → C 2×2 is analytic, where γ and ˜γ are the oriented contours shown in [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The jump contour ∪ 5 i=1Σi for the RH problem for T . where the branch is such that γ(ζ) is an analytic function of ζ ∈ C \ Σ5 and γ(ζ) ∼ 1 as ζ → ∞. Define also the function p : C \ Σ5 → C by p(ζ) = − r(ζ) 2πi Z Σ5 ln G(ξ) r+(ξ) dξ ξ − ζ , (2.16) where the branch for ln G is such that ln G(ζ) = ln F(isρ ζ + 1 2 ) − isρ (ln(s)ζ − h(ζ)) (2.17) with ln F defined as in (2.1). Outside small neighborhoods of b1… view at source ↗
Figure 4
Figure 4. Figure 4: The contour ΓR for the RH problem R. Furthermore, it was shown in [12, Section 4.3] that ∂s ln det 1 − K [PITH_FULL_IMAGE:figures/full_fig_p017_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The contours ˆσ and ˜σ for ζ /∈ Dδ/2(b1) ∪ Dδ/2(b2). Lemma 3.6. For any integer N ≥ 1, it holds that r(ζ) 2πi Z Σ5 DN (x(ξ)) r+(ξ)(ξ − ζ) dξ = O [PITH_FULL_IMAGE:figures/full_fig_p021_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The contour ˆσ for ζ ∈ Dδ/2(b1) ∪ Dδ/2(b2). O(s −ρ(2N+1)) as s → +∞ uniformly for ξ ∈ σˆ. Since r(ζ) ∼ ζ as ζ → ∞ and dist(ζ, σˆ) ≥ δ/2, we find    − r(ζ) 2πi Z σˆ DN (x(ξ)) r(ξ)(ξ − ζ) dξ = O(s −ρ(2N+1)), if ζ /∈ Dδ/2(b1) ∪ Dδ/2(b2) − r(ζ) 4πi Z ∂Dδ DN (x(ξ)) r(ξ)(ξ − ζ) dξ − r(ζ) 2πi Z σˆ\∂Dδ DN (x(ξ)) r(ξ)(ξ − ζ) dξ = O(s −ρ(2N+1)) if ζ ∈ Dδ/2(b1) ∪ Dδ/2(b2) as s → +∞ uniformly for ζ ∈ C\Σ5 and θ… view at source ↗
Figure 7
Figure 7. Figure 7: The contour L1. In particular, A1(ζ) is given explicitly by (3.4), the functions An(ζ) are holomorphic on C \ Σ5, satisfy An = O(ζ n) as ζ → ∞ uniformly for θ in compact subsets of (0, 1], and depend continuously on α and θ. Proof. Using that r+(ζ) + r−(ζ) = 0 for ζ ∈ Σ5, we obtain Z Σ5 dξ r+(ξ)(ξ − ζ) = 1 2 Z Σ5 dξ r+(ξ)(ξ − ζ) − 1 2 Z Σ5 dξ r−(ξ)(ξ − ζ) = 1 2 Z L dξ r(ξ)(ξ − ζ) , where L is a clockwise l… view at source ↗
Figure 8
Figure 8. Figure 8: The contour σK and the poles {ζj}∞ 0 of ψ( 1+α 2 −isρ ζ θ ) in the complex ζ-plane. The uppermost pole ζ0 lies a distance O(s −ρ ) from the origin as s → +∞. The horizontal line segment has a length of order O(1) as s → +∞ and crosses the imaginary axis half-way between the origin and ζ0. Using (6.5) and (6.6), we arrive at Tr[Y −1 − Y ′ −(J − I)σ3] = 1 2  Tr[(I − J)Y −1 + Y ′ +σ3] + Tr[Y −1 − Y ′ −(J − I… view at source ↗
read the original abstract

We consider the limiting process that arises at the hard edge of Muttalib--Borodin ensembles. This point process depends on $\theta > 0$ and has a kernel built out of Wright's generalized Bessel functions. In a recent paper, Claeys, Girotti and Stivigny have established first and second order asymptotics for large gap probabilities in these ensembles. These asymptotics take the form \begin{equation*} \mathbb{P}(\mbox{gap on } [0,s]) = C \exp \left( -a s^{2\rho} + b s^{\rho} + c \ln s \right) (1 + o(1)) \qquad \mbox{as }s \to + \infty, \end{equation*} where the constants $\rho$, $a$, and $b$ have been derived explicitly via a differential identity in $s$ and the analysis of a Riemann--Hilbert problem. Their method can be used to evaluate $c$ (with more efforts), but does not allow for the evaluation of $C$. In this work, we obtain expressions for the constants $c$ and $C$ by employing a differential identity in $\theta$. When $\theta$ is rational, we find that $C$ can be expressed in terms of Barnes' $G$-function. We also show that the asymptotic formula can be extended to all orders in $s$.

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

1 major / 1 minor

Summary. The manuscript considers the hard-edge limiting point process for Muttalib-Borodin ensembles with parameter θ > 0, whose kernel is built from Wright generalized Bessel functions. Building on the first- and second-order large-gap asymptotics of Claeys-Girotti-Stivigny, it derives explicit expressions for the constants c and C in the gap probability asymptotic P(gap on [0,s]) = C exp(−a s^{2ρ} + b s^ρ + c ln s) (1 + o(1)) by means of a differential identity in θ; when θ is rational it expresses C via Barnes' G-function, and it extends the expansion to all orders in s.

Significance. If the central justification holds, the work supplies explicit constants (including a closed-form expression in terms of the Barnes G-function for rational θ) and an all-order asymptotic for gap probabilities in this family of determinantal processes. The introduction of a θ-differential identity as an independent tool after the s-based analysis is a concrete methodological contribution that could extend to other parameter-dependent kernels; the explicit special-function results are falsifiable and strengthen the link between hard-edge statistics and classical special functions.

major comments (1)
  1. [Section deriving c and C from the θ-differential identity] The derivation of c and C (via the differential identity in θ applied to the limiting gap probability after the Claeys-Girotti-Stivigny s-asymptotics) requires an explicit argument that differentiation with respect to θ may be passed under the s → ∞ limit and that the resulting ODE on the limiting object is free of boundary terms affecting the constants. This step is load-bearing for the main claims on c and C and is not addressed by the first- and second-order analysis already in hand.
minor comments (1)
  1. [Abstract and the all-order extension paragraph] The abstract states that the asymptotic formula is extended to all orders, but the manuscript should clarify whether the all-order remainder is controlled uniformly in θ or only for fixed θ.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the need to strengthen the justification of the interchange between the θ-derivative and the large-s limit. We respond to the single major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Section deriving c and C from the θ-differential identity] The derivation of c and C (via the differential identity in θ applied to the limiting gap probability after the Claeys-Girotti-Stivigny s-asymptotics) requires an explicit argument that differentiation with respect to θ may be passed under the s → ∞ limit and that the resulting ODE on the limiting object is free of boundary terms affecting the constants. This step is load-bearing for the main claims on c and C and is not addressed by the first- and second-order analysis already in hand.

    Authors: We agree that an explicit justification for interchanging differentiation in θ with the s → ∞ limit is necessary and is not fully supplied by the existing s-asymptotics. In the revised manuscript we will add a dedicated paragraph (or short appendix) that supplies the missing argument. The justification will proceed from the analytic dependence on θ of the finite-s gap probability (inherited from the θ-dependence of the Muttalib–Borodin kernel) together with the uniform-in-θ convergence established by Claeys–Girotti–Stivigny on compact subsets of θ > 0; standard results on differentiation under the limit then apply. We will also verify directly that the resulting first-order ODE for the limiting gap probability introduces no additional boundary terms at s = ∞ that could modify the constants c and C. This addition will make the derivation of c and C self-contained. revision: yes

Circularity Check

0 steps flagged

No circularity: new differential identity in θ applied after prior asymptotics

full rationale

The derivation extends Claeys-Girotti-Stivigny first- and second-order results (obtained via s-differential identity and RH analysis) by introducing a separate θ-differential identity on the limiting kernel built from Wright generalized Bessel functions. No quoted step reduces a claimed constant c or C to a fitted parameter or to the same RH problem by construction; the θ-identity is invoked as an independent tool whose justification is external to the s-asymptotics already established. Self-citations are absent from the load-bearing chain, and the extension to all orders in s follows from the same non-circular limiting process. This is the standard case of an independent analytic continuation rather than definitional equivalence.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the validity of the θ-differential identity and on the prior first- and second-order asymptotics of Claeys et al.; no free parameters are introduced in the abstract, but the method implicitly assumes analytic continuation in θ is justified for the limiting kernel.

axioms (1)
  • domain assumption The limiting kernel built from Wright's generalized Bessel functions admits a differential identity in θ that can be used to evaluate the gap-probability constants after the s-asymptotics are known.
    Invoked to obtain c and C; location not visible in abstract.

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