Asymptotics for a class of planar orthogonal polynomials and truncated unitary matrices

Alfredo Dea\~no; Kenneth T-R McLaughlin; Leslie Molag; Nick Simm

REVIEW 3 major objections 4 minor 28 references

Asymptotics for a class of planar orthogonal polynomials and truncated unitary matrices

T0 review · 3 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read This paper proves explicit asymptotics for planar orthogonal polynomials and derives the full moment expansion and central limit theorem for |det(B_n-x)| of truncated unitary matrices.

desk verdict A strong, likely correct Riemann–Hilbert paper with one real but fillable gap in Lemma 2.1 and a definite typo in the CLT proof; worth a serious referee. read the letter →

arxiv 2505.12633 v2 pith:SD2O7EZL submitted 2025-05-19 math-ph math.CAmath.CVmath.MPmath.PR

Alfredo Deaño , Kenneth T-R McLaughlin , Leslie Molag , Nick Simm This is my paper · ORCID

classification math-phmath.CAmath.CVmath.MPmath.PR MSC 33C4533E1760B2041A60

keywords planarorthogonalpolynomialstruncatedunitarymatricesRiemann-Hilbertproblemstrongnon-unitaritycharacteristicpolynomialmomentscentrallimittheoremsigma-PainleveVBarnesG-function

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 establishes the large-n behaviour of planar orthogonal polynomials for the weight (1-|z|^2)^{$\alpha$-1}|z-x|^gamma on the unit disc, in the regime where $\alpha$ grows linearly with n. From this it derives an explicit asymptotic formula for E(|det(B_n-x)|^gamma) for truncated Haar unitary matrices in the strong non-unitarity regime mu=n/N -> tilde mu in (0,1). The formula has relative error O(1/n) and yields a central limit theorem for log|det(B_n-x)| whose variance grows like (1/4)log n. The proof converts planar orthogonality into contour orthogonality and applies steepest descent analysis to the associated Riemann-Hilbert problem, avoiding the need for local parametrices.

What carries the argument

The key step is Lemma 2.1: planar orthogonality with weight (1-|z|^2)^{$\alpha$-1}|z-x|^gamma is equivalent to contour orthogonality with weight w(z)=(1-$x^{2}$ z)^{$\alpha$+gamma/2}((z-1)/z)^{gamma/2} on a loop enclosing [0,1]. This contour orthogonality is encoded in a 2x2 Riemann-Hilbert problem. A g-function built from the potential V(z)=log z - c log(1-$x^{2}$ z), with level curves Re phi = constant, produces normalized jumps; after lens opening and a global parametrix, the remaining Riemann-Hilbert problem is small-norm, so the orthogonal polynomials are expressed as contour integrals. These integrals are evaluated through an incomplete gamma function identity, yielding explicit three-region asymptotics without constructing a local parametrix. A differential identity for d/dx log R_gamma(x) in terms of the Riemann-Hilbert matrix entries, together with the explicit radially symmetric value at x=0, completes the moment formula.

What would settle it

Simulate truncated unitary matrices for moderate n, for example n=50 to 500 with mu=1/2, x=0.5, and gamma=1, and compare the empirical E(|det(B_n-x)|^gamma) with the theorem's leading term, checking that the relative error decays like O(1/n). Separately, for gamma in (-2,0), numerically compare the planar integral defining the orthogonality in (1.2) with the contour integral in (2.2) for low-degree polynomials to verify Lemma 2.1 directly.

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Extended reading notes

Core claim

The central claim is that in the strong non-unitarity regime, with fixed gamma satisfying Re gamma > -2 and fixed x in [0,$\sqrt$(mu)), the moments satisfy E(|det(B_n-x)|^gamma) = $n^{{gamma^2/8}}$ $mu^{{gamma n/2}}$ ((1-mu)/(1-$x^{2}$))^{$\alpha$ gamma/2} C_{gamma,mu}(x)(1+O(1/n)) with C_{gamma,mu}(x) explicitly given in terms of the Barnes G-function. This is proved through uniform asymptotics of the monic planar orthogonal polynomials in the exterior, interior, and near z=1; near z=1 the behaviour is governed by an incomplete gamma function. A corollary is the convergence in distribution of (log|det(B_n-x)| - kappa_1/2)/((1/2)$\sqrt$(log n)) to a standard normal, where kappa_1 is the explicit centering term. In a separate double-scaling regime $x^{2}$=1-v/n with $\alpha$ fixed, the same quantity is expressed through a $\sigma$-Painleve V function following the theory of Toeplitz determinants with Fisher-Hartwig singularities.

Load-bearing premise

The whole construction rests on Lemma 2.1, the reduction of planar orthogonality to contour orthogonality; for the singular range gamma in (-2,0) the Green's theorem step near z=x is stated but the details are omitted, so if that reduction fails, the later theorems lose their foundation.

Editorial extensions

If this is right

The moment generating function of log|det(B_n-x)| now has a full leading-order expansion, so cumulants of the logarithm of the characteristic polynomial can be extracted order by order in the strong non-unitarity limit.
A central limit theorem holds for log|det(B_n-x)| for every fixed x in the bulk of the limiting spectrum, generalizing the previously known determinant case x=0.
The zeros of the planar orthogonal polynomials accumulate on the explicit level curve Gamma_1, and the polynomials have different algebraic, rational, and incomplete-gamma behaviours in the exterior, interior, and near-boundary regions.
In the double-scaling weak regime, the moments are expressed in terms of a sigma-Painleve V function, giving a concrete connection between truncated unitary matrices and the Fisher-Hartwig/Toeplitz theory.
The integer moment case gamma=2k reproduces results previously obtained by duality arguments, providing a consistency check for the new Riemann-Hilbert approach.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

One could test whether the same parametrix-free steepest descent scheme applies to planar weights with several point charges or with other radially symmetric base weights, which would give complete asymptotics beyond the Gaussian and unit-disc cases.
The variance (1/4)log n seen in the central limit theorem is the signature of a log-correlated field; comparing higher cumulants with Gaussian multiplicative chaos predictions would be a natural numerical check.
The differential identity used here might be iterated to compute subleading coefficients in the O(1/n) correction, and it may connect the strong-regime formula to the sigma-Painleve V expression across the critical curve x=sqrt(mu).

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Quick take: this paper is solid and deserves a real referee. The main asymptotics are very likely correct, and the one explicit gap—Lemma 2.1 for γ ∈ (−2,0)—is real but fillable, not fatal. There is also a small but definite typo in the CLT proof that should be fixed.

What is new: the measure (1−|z|²)^{α−1}|z−x|^γ on the disc is a genuine generalization of the Gaussian weights analyzed in [22] and [26], and the paper gets uniform asymptotics for the planar orthogonal polynomials in every region of the plane, then converts them into the leading asymptotics (1.10) for moments of characteristic polynomials of truncated unitary matrices in the strong non-unitarity regime. The method is a real departure: instead of building a local parametrix at the branch point z=1, the contour is steered through a regular point and the resulting RHP is handled with a small-norm Neumann series. That trick works and is one of the nice contributions. The Painlevé V connection in the double-scaling regime is correctly imported from [14] and adds value.

The steepest descent analysis is careful. The g-function, lens opening, global parametrix, and error estimates are written out with explicit exponents, and Section 3.5 gives a substantially detailed small-norm argument, not just a hand-wave. The differential identity in Appendix B is in good shape.

Soft spots, in proportion. First, Lemma 2.1 is the hinge: every later theorem depends on the equivalence of planar and contour orthogonality. For γ ∈ (−2,0), the proof stops at “we omit the details” exactly at the singular term near z=x. A quick estimate shows the small-circle boundary contribution behaves like ε^{γ+k+2}→0, so the gap is very plausibly closed, but it should be closed in print rather than deferred. Second, the proof of Corollary 1.6 says the CLT follows by taking γ = t/(2√log n), which is off by a factor of 4; the substitution should be γ = 2t/√log n. This looks like a typo, not a conceptual error, but as written the proof gives the wrong MGF limit. Third, the uniformity claim in Theorem 1.4 is argued a little briskly; the arc-length estimates for γ=0 are fine and the continuation to Re γ > −2 is plausible, but a referee will want the constants tracked more carefully.

Citation pattern is fine. The reuse of [15] is legitimate because the contour-orthogonality reduction is genuinely from that earlier paper, and the external Painlevé V results are cited precisely.

Who this is for: people working on planar orthogonal polynomials, Riemann–Hilbert asymptotics, and characteristic polynomials of non-Hermitian or truncated unitary ensembles. I would send it to a serious referee; a desk rejection would be wrong.

Referee Report

3 major / 4 minor

Summary. The paper develops the large-n asymptotics of planar orthogonal polynomials for the measure dµ(z)=(1-|z|^2)^{α-1}|z-x|^γ 1_{|z|<1} d^2z in the strong non-unitarity regime n/N → µ̃∈(0,1). The authors convert the planar orthogonality into a contour orthogonality, set up a Riemann-Hilbert problem, and then apply Deift-Zhou steepest descent. The main results are: all-region asymptotics for the (rescaled) monic orthogonal polynomials (Theorem 1.1), an asymptotic expansion for the moments E(|det(B_n-x)|^γ) with explicit constants and uniform error O(1/n) (Theorem 1.4), a central limit theorem for log|det(B_n-x)| (Corollary 1.6), and a double-scaling result near x=1 expressed through a σ-Painlevé V solution (Theorem 1.8).

Significance. If the technical gaps are completed, this is a substantial contribution. The paper gives the first all-region asymptotic description for this non-Gaussian planar ensemble with a point charge, and it does so with explicit, parameter-free constants and no fitted parameters. The resulting moment formula and CLT for truncated unitary matrices are concrete and independently checkable. The method also avoids local parametrices in the strong regime, which is a useful structural improvement over earlier Gaussian-weight treatments, and the weak-regime Painlevé V connection extends the Toeplitz-determinant results of Claeys–Its–Krasovsky to the truncated-unitary partition function. The authors are explicit about error terms and about the range of parameters claimed, and they correctly identify the singular case γ∈(-2,0) as requiring separate treatment.

major comments (3)

[Appendix A, Lemma 2.1] The proof of Lemma 2.1 explicitly omits the singular case γ∈(-2,0). After equation (A.4) the text states: 'for γ∈(-2,0) there is singular behavior near z=x, but one can still verify the statement. We omit the details.' This is a load-bearing gap, because Theorems 1.1 and 1.4 both claim uniform validity for Reγ>-2, which includes this interval. The Green's-theorem reduction requires a justification that the integration-by-parts step applies to the unbounded integrand p_j(z)(z-x)^{γ/2}h_k(z, \bar z, x) near z=x, and a proof that the small-circle boundary contribution around x vanishes. Since the contour orthogonality (2.2), the Riemann-Hilbert problem (2.10)-(2.13), and all subsequent steepest-descent analysis depend on Lemma 2.1, this omission affects the foundation of the paper. Please supply the missing limiting argument for -2<γ<0, or explicitly restrict the main theorems to Reγ>0 if the singular range is not intended to be covered.
[§3.5, equation (3.23)] The expansion (3.23) for R(z) is only sketched. The Neumann series (3.33)-(3.34) is written down, but the step 'expanding up to order 1 and bounding the remaining terms' is not supported by explicit estimates for the iterated Cauchy operators C_{Σ_{r1}}, C_{Σ_{r2}}. The case r=r2=z0 is also delicate: there φ_r(r2)=0 and the RHP is not directly in the small-norm setting; the rescaling R_ε is mentioned but its analysis is not provided. The later estimates in §3.6-3.7, Proposition 3.4, and ultimately Theorem 1.1 rely on (3.36) and the claimed exponential smallness of the error. A complete proof of (3.23) with explicit norm bounds is therefore needed.
[§4, proof of Theorem 1.4] The proof of uniformity in Theorem 1.4 is not fully justified. After treating γ=0 in (4.5)-(4.7), the text says 'The general case Reγ>-2 is essentially the same' and asserts that the added factors depend continuously on x and γ. This skips the behavior of the error as γ approaches -2 and as x approaches √µ̃-δ, exactly where the exponent difference φ(r1)-φ(z0) degenerates and the factors of h_γ grow. Since the theorem claims uniformity over compact subsets of Reγ>-2 and over [0,√µ̃-δ], the omitted continuity argument is load-bearing. Please give the explicit dependence of the error term in (4.7) on γ and x, or state uniformity only over fixed compact subsets of (γ,x) that avoid the degenerate limits.

minor comments (4)

[§3.6, equation (3.42)] The error term O(e^{k(2φ(t)-φ(z0))}) appears to contain an undefined constant k; presumably this should be n, matching the rest of the section. Please clarify.
[§4] There is a typo: 'preceeding' should be 'preceding'.
[Theorem 1.4] The theorem states uniformity for 0≤x<√µ̃-δ, while the measure (1.1) and the polynomial construction are stated for x>0 in the introduction. The case x=0 is computed separately in the proof and in Remark 1.3, but the theorem statement should make explicit whether x=0 is included in the uniform statement.
[Lemma 3.2] The domain of the coefficient functions c_m(z) is written as C\{γ_t}; this should presumably be C\γ_t, i.e. the complement of the curve, not a set minus a set of points.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular reduction found; the only flagged weakness is an omitted singular-case verification in Lemma 2.1, which is a proof gap rather than circularity.

full rationale

The derivation chain is: planar orthogonality (1.2) → Lemma 2.1 contour orthogonality (2.2) with weight (2.1) → Riemann-Hilbert problem (2.10)–(2.13) → steepest descent analysis yielding Theorem 1.1 → differential identity (2.15) → integration from x=0 using the exact radial value R_γ(0) to obtain Theorem 1.4. Each link is explicit and parameter-free. The weight (2.1) is obtained by algebraically rewriting the planar integral in Appendix A, not by assuming the asymptotics it feeds. The RHP solution (2.13) is written in terms of the same orthogonal polynomials, and the small-norm/steepest-descent estimates depend only on the analytic properties of g and φ. The constant R_γ(0) is computed exactly from the rotationally symmetric case x=0, and no fitted constant is introduced. External inputs such as [14], [16], [25], and [26] are independent results whose assumptions do not include the strong-regime theorem proved here. Self-citations to [15] are used for a similar contour reduction and for product asymptotics, but Lemma 2.1's proof is reproduced in Appendix A and the product step is standard Barnes-G asymptotics, so these citations are not load-bearing circularity. One flagged passage is Appendix A, after Eq. (A.4): 'for γ ∈ (−2, 0) there is singular behavior near z = x, but one can still verify the statement. We omit the details.' This is a genuine completeness gap in a load-bearing lemma and a correctness risk for part of Re γ > −2, but it is not circularity: the claimed reduction is Green's theorem plus algebraic manipulation, and it does not presuppose Theorem 1.1 or Theorem 1.4. Accordingly, no significant circularity is found.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new entities and fits no free parameters; alpha, gamma, x and mu~ are inputs. The load-bearing external inputs are the contour-orthogonality technique (self-cited [15]) and the Painleve V theory of [14].

assumptions (4)

domain assumption Contour orthogonality reduction (Lemma 2.1) is valid for all Re gamma > -2, including gamma in (-2,0), where the Green's theorem step near z=x is asserted without proof.
The full RHP formulation rests on this lemma; the paper explicitly omits the singular-case verification in Appendix A.
standard math The Deift-Zhou steepest descent method applies to the RHP with the g-function and level curves Gamma_r of Lemma 3.1.
The paper proves the needed conformal properties of phi and Gamma_r in Lemma C.1; the steepest descent framework is standard.
standard math The RHP (2.10)-(2.12) has a unique solution with the prescribed behavior at z=1.
Standard existence and uniqueness for orthogonal polynomial RHPs; boundedness assumptions are imposed.
standard math Toeplitz determinant asymptotics of Claeys-Its-Krasovsky [14] apply to the symbol (5.3) with a=(gamma+alpha)/2, b=alpha/2, for the stated parameter ranges.
Theorem 1.8 and Proposition 5.2 inherit the results of [14, Theorems 1.1 and 1.4].

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Pith. "Pith review of Asymptotics for a class of planar orthogonal polynomials and truncated unitary matrices." pith.science (2026). https://pith.science/paper/SD2O7EZL

@misc{pith2026250512633,
  author       = {Pith},
  title        = {Pith review of: Asymptotics for a class of planar orthogonal polynomials and truncated unitary matrices},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SD2O7EZL}},
  note         = {Machine review of arXiv:2505.12633}
}

abstract

We carry out the asymptotic analysis as $n \to \infty$ of a class of orthogonal polynomials $p_{n}(z)$ of degree $n$, defined with respect to the planar measure \begin{equation*} d\mu(z) = (1-|z|^{2})^{\alpha-1}|z-x|^{\gamma}\mathbf{1}_{|z| < 1}d^{2}z, \end{equation*} where $d^{2}z$ is the two dimensional area measure, $\alpha$ is a parameter that can grow with $n$, while $\gamma>-2$ and $x>0$ are fixed. This measure arises naturally in the study of characteristic polynomials of non-Hermitian ensembles and generalises the example of a Gaussian weight that was recently studied by several authors. We obtain asymptotics in all regions of the complex plane and via an appropriate differential identity, we obtain the asymptotic expansion of the partition function. The main approach is to convert the planar orthogonality to one defined on suitable contours in the complex plane. Then the asymptotic analysis is performed using the Deift-Zhou steepest descent method for the associated Riemann-Hilbert problem.

Figures

Figures reproduced from arXiv: 2505.12633 by the authors.

**Figure 1.** The zeros of the orthogonal polynomials pn(xz) as defined in (1.2). Here N = 2n = 75, γ = 1 2 and x = 7 12 . The curve on the left is Γ1. The curve on the right is another component of the level curve φ(z) := |z||1 − x 2 z| = 1 − x 2 (note that ˜µ = 1 2 ). The point z0 = ˜µx−2 = 72 49 is the only point where two level curves of φ meet and it is a saddle point. x = 0 is trivial since the rotational symmetry gives pn(… view at source ↗

**Figure 2.** Curve C and branch cut for the orthogonal polynomials pk(z). Lemma 2.1. The planar orthogonal polynomials p˜j (z) = x −jpj (xz) introduced in (1.2) are characterized by the following contour orthogonality (2.2) I C p˜j (z)z −kw(z) dz 2iz = δj,k 1 χˆj , k = 0, . . . , j. where (2.3) ˆχj = Γ( γ 2 + j + 1)Γ(α) Γ( γ 2 + j + α + 1) χj . 8 [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** Graphic representation of the level curve Γ˜ r as defined in (3.7), here for r ̸= z0 = 1 (c+1)x2 . Lemma 3.1. Let c > 0 and 0 < r ≤ z0. Then Γ˜ r as defined in (3.7) consists of two closed Jordan curves, which both intersect the real line in exactly two points. They can only intersect eachother in the case r = z0, in which case z0 is their only point of intersection. One of these Jordan curves, Γr, encloses the inte… view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: The image under ζ = ϕ(z) of the sets S1 \ (−∞, 0), S2, S ∗ 2 , S3 and S ∗ 3 from Lemma C.1, which consists of horizontal strips. The inverse image of the vertical line segment with Re ζ = ϕ(r) is given by the two Jordan curves in [PITH_FULL_IMAGE:figures/full_fig_p014…

**Figure 5.** Figure 5: Contour ΓS, for the case of strict inequalities 1 < r1 < r < r2 < x−2 . • On ΣS, the matrix S(z) has jumps (3.16) S+(z) = S−(z)    [PITH_FULL_IMAGE:figures/full_fig_p017_5.png]

**Figure 6.** Figure 6: Schematic representation of the sets S1, S2, S ∗ 2 , S3 and S ∗ 3 from Lemma C.1. ∂S2 ∩ ∂S3 and ∂S ∗ 2 ∩ ∂S ∗ 3 are mapped to [e ϕ(z0) , ∞). [0, z0] and [z0, x−2 ] are mapped to [0, eϕ(z0) ]. ∂S1 and [x −2 , ∞) correspond to arg ζ = ±πc. For cases (ii) and (iii), note …

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

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S-Y. Lee and M. Yang. Discontinuity in the asymptotic behavior of planar orthogo- nal polynomials under a perturbation of the Gaussian weight. Comm. Math. Phys. , 355(1):303–338, 2017

work page 2017

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Claeys, A

T. Claeys, A. Its, and I. Krasovsky. Emergence of a singularity for Toeplitz determi- nants and Painlev´ e V.Duke Math. J. , 160(2):207 – 262, 2011. 41

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