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Hard edge asymptotics of correlation functions between singular values and eigenvalues

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Pith's one-line read Universal hard-edge limit found for correlations between one eigenvalue and many singular values

desk verdict New large-n limits for eigenvalue–singular-value correlations, with an honest gap in the general Pólya-ensemble proof. read the letter →

arxiv 2501.15765 v3 pith:PBV2DMZ7 submitted 2025-01-27 math.PR math-phmath.MPmath.STstat.TH

Matthias Allard This is my paper · ORCID
classification math.PRmath-phmath.MPmath.STstat.TH MSC 60B2015B52
keywords singularvalueseigenvaluesbi-unitarilyinvariantensemblespolynomialensemblePólya1k-pointcorrelationfunctionk-cross-covariancehardedgescaling
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

This paper derives the large-matrix-size limit of the joint statistics of one eigenvalue radius and k singular values at the hard edge (near zero) for a large class of random complex matrices called polynomial ensembles, including all Pólya ensembles such as Laguerre, Jacobi, Cauchy-Lorentz, and Muttalib-Borodin ensembles. The main result is an explicit pointwise limit formula for the scaled 1,k-point function, expressed solely in terms of the scaled correlation kernel of the singular values. A surprising universal feature emerges: the smallest squared eigenradius scales like 1/n relative to the smallest squared singular values, so the two kinds of values live on different microscopic scales. The paper also gives an explicit limit at the 'soft-hard' edge of Jacobi ensembles, where the singular values have an upper hard edge but the eigenradii see a soft edge.

What carries the argument

The central object is the scaled correlation kernel $K^{{(∞)}}$(x,y) = lim_{n→∞} (1/(nν_n)) K_n(x/(nν_n), y/(nν_n)) of the determinantal point process on squared singular values, together with the explicit limit $φ^{{(∞)}}$(x,t)=x(t+x−1)$e^{{−x}}$$e^{{−t}}$ of the function φ_n that encodes the coupling between eigenradius and singular values. For Pólya ensembles, the kernel is built from biorthonormal functions p_j, q_j, and the assumptions are rephrased directly on the Mellin transform of the Pólya weight w_n; the limiting kernel then admits an integral representation in terms of $Q^{{(∞)}}$ and $P^{{(∞)}}$ (which reduce to Bessel functions when the scaled weight is Laguerre-like). The proof machinery is dominated convergence, with careful splitting of the integration domain into a compact part where limits pass and a non-compact remainder that vanishes in the large-n, large-R limits.

What would settle it

Find a Pólya ensemble that satisfies Assumptions 2.8 but for which no choice of biorthonormal functions satisfies the exponential bounds (2.8)-(2.9); then the claimed implication from Assumptions 2.8 to Theorem 2.12 collapses. Concretely, one could test a weight w_n(x) = $x^{{α_n}}$$e^{{-x}}$ with α_n growing with n such that the Mellin transform has poles in the strip (1,n), and check whether the scaled 1,1-point function still converges to the formula (2.14).

Watch

Extended reading notes

Core claim

The paper establishes that, under assumptions guaranteeing a well-behaved scaled kernel, the 1,k-point correlation function between one squared eigenradius and k squared singular values has the pointwise limit n/$ν_n^{{k+1}}$ f_{1,k}(r/ν_n; a_1/(nν_n), ..., a_k/(nν_n)) → ∫∫ (dv/v) $φ^{{(∞)}}$(v/r, t) det[...], with $φ^{{(∞)}}$(x,t)=x(t+x−1)$e^{{−x}}$$e^{{−t}}$ and the determinant built from the scaled limiting kernel $K^{{(∞)}}$. The same limit holds for all polynomial ensembles whose limiting kernel coincides, so Laguerre, Jacobi, and Cauchy-Lorentz ensembles share identical hard-edge asymptotics. For Jacobi ensembles, the paper also derives the soft-hard edge limit, where the limiting cross-covariance factorizes as a Gaussian in the eigenradius times a Bessel-kernel determinant in the singular values.

Load-bearing premise

The whole result rests on the assumption that the scaled correlation kernel $K^{{(∞)}}$ exists and that the biorthonormal functions p_j, q_j satisfy n-independent exponential bounds of the form (2.8)-(2.10), which the paper verifies for concrete ensembles but does not prove in full generality for all polynomial ensembles.

Editorial extensions

If this is right

  • If the central claim is correct, the 1/n scaling ratio between the eigenradius scale and the singular-value scale at the hard edge is universal for polynomial ensembles, meaning the smallest eigenradius is typically n times farther from zero than the smallest singular values.
  • All ensembles with the same scaled limiting kernel share the same limiting 1,k-point function and 1,k-cross-covariance at the hard edge; the paper shows this explicitly for Laguerre, Jacobi, and Cauchy-Lorentz ensembles.
  • For Muttalib-Borodin ensembles, the explicit limiting kernel and the compact-integral form of Corollary 2.18 provide a numerically efficient way to compute the limiting 1,k-point function and cross-covariance.
  • At the Jacobi soft-hard edge, the limiting cross-covariance is a factor 1/n smaller than the 1,k-point function, so it acts as a first-order correction term that vanishes relative to the product of one-point functions.
  • The Gaussian dependence on the eigenradius at the soft-hard edge links the cross-covariance to the derivative of the limiting eigenradius density, a structure previously observed in edge corrections for β-ensembles.

Reading between the lines

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

  • Beyond the paper: the universal 1/n scaling ratio may persist for bi-unitarily invariant ensembles that are not polynomial, but the proof route used here would need a different argument because the kernel limit may not be a function; a direct numerical test on a non-polynomial bi-unitarily invariant ensemble would clarify this.
  • Beyond the paper: the soft-hard edge result suggests a general principle: when eigenradii and singular values do not share the same hard wall, their cross-correlations decay faster (here by an extra 1/n) than when they share the origin edge; this could be tested on other two-edge ensembles.
  • Beyond the paper: the explicit formula for the limiting cross-covariance could be used to estimate finite-n corrections to eigenvalue-singular-value independence, which may be relevant for non-Hermitian quantum chaos models where singular values are the effective spectral observable.
  • Beyond the paper: since Assumptions 2.8 translate directly into Mellin-transform bounds, one could algorithmically check whether any given Pólya weight satisfies them, yielding a practical criterion for when the hard-edge limit applies to product ensembles formed by multiplicative convolution.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper studies the large-n limit of the joint density of one squared eigenradius and k squared singular values (the 1,k-point function) for bi-unitarily invariant random matrix ensembles whose singular-value law is a polynomial ensemble. The main results are: (i) Theorem 2.5, which gives a hard-edge scaling limit at the origin in terms of a double integral over the limiting kernel, under abstract bounds on the bi-orthonormal system (Assumptions 2.1); (ii) Theorem 2.12 and Corollary 2.18, which provide more explicit formulas for Pólya ensembles under Mellin-transform conditions (Assumptions 2.8), including an application to Muttalib–Borodin ensembles; and (iii) Theorem 3.2, a soft-hard edge limit for Jacobi ensembles. The paper also proves the consistency of some of the assumptions in Appendix A and discusses the universal scaling ratio of 1/n between the eigenradius and singular-value scales.

Significance. If the technical assumptions are satisfied, this is a substantial extension of the finite-n formulas of [3] to the hard-edge scaling regime, giving universal 1,k-point correlation limits that couple eigenradii with singular values for a wide class of bi-unitarily invariant ensembles. The explicit formulas for Muttalib–Borodin ensembles and the Jacobi soft-hard edge are new and potentially useful for applications in non-Hermitian quantum chaos and QCD. The paper is written carefully, with detailed proofs and a consistency appendix, and it makes no use of fitted parameters. The main weakness is that the bridge from the readily checkable Pólya assumptions to the abstract bi-orthonormal bounds is not fully proven; the author acknowledges this gap, which currently makes Theorem 2.12 not fully established as stated.

major comments (2)
  1. [§2.2, Remark 2.11 and Theorem 2.12]
  2. [§4.2, proof of Corollary 2.18]
minor comments (5)
  1. [§4.1, proof of Lemma 4.1]
  2. [§2.1, Assumptions 2.1(3)]
  3. [§2.2, Proposition 2.13]
  4. [§5.1, proof of Theorem 3.2]
  5. [Figure 1]

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the large-n limits are derived from a prior finite-n formula and kernel asymptotics; no fitted quantity is renamed as a prediction and no limit reduces to its own assumptions by construction. The admitted gap from Assumptions 2.8 to (2.8) is a proof gap, not circularity.

full rationale

The derivation chain is: Theorem 1.1 and Proposition 1.2, quoted from the author's earlier paper [3], give finite-n formulas for the 1,k-point function and the 1,k-cross-covariance. The paper then chooses scalings, assumes the kernel limit (2.7) and the bounds (2.8)-(2.10), and applies dominated convergence to obtain Theorem 2.5. Formula (2.14) is exactly the formal limit of (1.16) under Assumptions 2.1; the limiting kernel K^(∞) is an assumed input, not a quantity fitted to f_{1,k}, so no prediction is forced by construction. For Pólya ensembles, Assumptions 2.8 are conditions on Mellin transforms and weights, and Proposition 2.13 (adapted from [27], by a different author) converts them into K^(∞); the paper verifies those conditions for Laguerre, Jacobi, Cauchy-Lorentz and Muttalib-Borodin ensembles rather than using the target f_{1,k} itself. The only substantial self-citation is [3], which supplies the finite-n starting point; the target hard-edge limit is not contained in [3], so the central claim has independent content. The paper itself flags in Remark 2.11 that Assumption (2.26) 'is likely to imply (2.8), yet it is neither clear nor shown,' and in Section 6 it says it 'believe[s] that Assumptions 2.8 strictly imply Assumptions 2.1.' This is an honest identification of an unproved technical bridge used in Corollary 2.18 and Theorem 2.12. It affects proof completeness for general Pólya ensembles, but it is not circular: no equation is reused as its own conclusion, and no parameter estimated on a subset is later 'predicted.' The soft-hard edge result for Jacobi ensembles is derived independently from classical Jacobi-polynomial asymptotics and the same finite-n formula, again without fitted inputs. I therefore find no circular step and assign score 0; the admitted gap belongs to correctness risk, not circularity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No parameters are fitted to data; the ensemble parameters α, β, θ are fixed inputs. The scaling sequences ν_n, ξ_n are chosen to make limits exist, not fitted. The results rely on prior work for the finite-n formula and on technical assumptions about the scaled kernels and Mellin transforms.

assumptions (5)
  • domain assumption Assumptions 2.1: existence of the point-wise limit K^(∞) in (2.7) and the n-independent bounds (2.8), (2.9), (2.10) on the bi-orthonormal functions p_j, q_j.
    Theorem 2.5 is conditional on these bounds, which license interchanging limit and integrals. They are checked for the named examples but not proven for all polynomial ensembles.
  • domain assumption Assumptions 2.8: the Mellin transform of the Pólya weight is holomorphic on the stated domain and satisfies the scaling limits (2.24) and bounds (2.25)-(2.28).
    Theorems 2.12, Corollary 2.18 and Proposition 2.13 rely on these conditions. The paper notes the implication from (2.26) to (2.8) is not shown.
  • standard math Finite-n 1,k-point formula (1.16) and cross-covariance formula (1.21) from [3] for bi-unitarily invariant ensembles with polynomial/Pólya singular value densities.
    The asymptotic results start from these fixed-n expressions; their correctness is taken from the prior publication.
  • standard math Integral representation of the Pólya kernel (2.40) and the explicit bi-orthonormal functions (2.41) from [28].
    Used in Corollary 2.18 and Proposition 2.13 to compute the scaling limit of the kernel.
  • standard math Asymptotic expansions for Jacobi polynomials outside and near the orthogonality interval (Propositions 5.3 and 5.4 from Szegő).
    Used in the Laplace-method proof of the soft-hard edge limit for Jacobi ensembles.

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Pith. "Pith review of Hard edge asymptotics of correlation functions between singular values and eigenvalues." pith.science (2026). https://pith.science/paper/PBV2DMZ7

@misc{pith2026250115765,
  author       = {Pith},
  title        = {Pith review of: Hard edge asymptotics of correlation functions between singular values and eigenvalues},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PBV2DMZ7}},
  note         = {Machine review of arXiv:2501.15765}
}
abstract

Any square complex matrix of size $n\times n$ can be partially characterized by its $n$ eigenvalues and/or $n$ singular values. While no one-to-one correspondence exists between those two kinds of values on a deterministic level, for random complex matrices drawn from a bi-unitarily invariant ensemble, a bijection exists between the underlying singular value ensemble and the corresponding eigenvalue ensemble. This enabled the recent finding of an explicit formula for the joint probability density between $1$ eigenvalue and $k$ singular values, coined $1,k$-point function. We derive here the large $n$ asymptotic of the $1,k$-point function around the origin (hard edge) for a large subclass of bi-unitarily invariant ensembles called polynomial ensembles and its subclass P\'olya ensembles. This latter subclass contains all Meijer-G ensembles and, in particular, Muttalib-Borodin ensembles and the classical Wishart-Laguerre (complex Ginibre), Jacobi (truncated unitary), Cauchy-Lorentz ensembles. We show that the latter three ensembles share the same asymptotic of the $1,k$-point function around the origin. In the case of Jacobi ensembles, there exists another hard edge for the singular values, namely the upper edge of their support, which corresponds to a soft edge for the eigenvalue (soft-hard edge). We give the explicit large $n$ asymptotic of the $1,k$-point function around this soft-hard edge.

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