arxiv: 2604.24747 · v1 · submitted 2026-04-27 · 🧮 math.CA · math.PR

Recognition: unknown

A determinant identity for the sum of contour integral matrices

Tejaswi Tripathi, Zhipeng Liu

Pith reviewed 2026-05-07 17:37 UTC · model grok-4.3

classification 🧮 math.CA math.PR

keywords determinant identitycontour integralsFredholm determinantmatrix sumintegral kernelcomplex analysis

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The pith

The determinant of the sum of two contour-integral matrices equals a Fredholm determinant det(I + K) on one contour.

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

The paper derives an identity that rewrites the determinant of the sum of two finite matrices, each with entries built from contour integrals, as the Fredholm determinant of an integral operator. One matrix uses a closed contour integral over a circle involving functions p and f, while the other uses an integral over a different contour Γ with functions q and g. Under suitable assumptions on these functions and the contours, the identity converts det(A + B) directly into det(I + K) where K acts on Γ. This provides a bridge from a discrete n by n determinant to a continuous operator on the second contour and extends an earlier identity in the literature.

Core claim

We consider the matrices A and B with entries A(i,j) = (1/(2πi)) ∮_0 z^{i-j-1} p_i(z) f_j(z) dz and B(i,j) = (1/(2πi)) ∫_Γ q_i(z) g_j(z) dz. The central claim is that det(A + B) equals the Fredholm determinant det(I + K), where K is an integral kernel acting on the contour Γ. This holds under suitable assumptions on the functions p, q, f, g and the contours, generalizing a recent identity from the cited work.

What carries the argument

The Fredholm determinant det(I + K) of an integral operator with kernel K on contour Γ, which equals det(A + B) for the contour-integral matrices A and B.

If this is right

The finite determinant det(A + B) can be analyzed using the spectral properties of the integral operator with kernel K on Γ.
The identity extends the earlier result to broader classes of functions p, q, f, g.
Limits or asymptotics of det(A + B) as n grows can be studied by passing to the Fredholm operator on the contour.

Where Pith is reading between the lines

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

The reduction may simplify large-n analysis by replacing matrix computations with operator-theoretic tools that admit asymptotic expansions.
Similar contour-integral representations could yield analogous identities for other families of matrices arising in integrable systems or orthogonal polynomial ensembles.
Numerical approximation of the Fredholm determinant might prove more stable than direct evaluation of the finite matrix for moderate to large n.

Load-bearing premise

The functions p, q, f, g and contours must satisfy conditions ensuring the integrals define the matrices properly and the kernel K yields a well-defined Fredholm determinant.

What would settle it

Pick explicit analytic functions p_i, q_i, f_j, g_j and contours satisfying the assumptions, compute the finite matrix det(A + B) directly for small n, construct the corresponding kernel K, and check whether the numerically evaluated Fredholm det(I + K) matches.

read the original abstract

We derive an identity for the determinant of the sum of two $n\times n$ matrices, $A$ and $B$, whose entries are defined via contour integrals. Specifically, we consider $A(i,j)=\frac{1}{2\pi\mathrm{i}}\oint_0 z^{i-j-1}p_i(z)f_j(z)\mathrm{d} z$ and $B(i,j)= \frac{1}{2\pi\mathrm{i}}\int_{\Gamma} q_i(z)g_j(z) \mathrm{d} z$. Under suitable assumptions on the functions $p,q,f,g$, we show that $\det(A+B)$ can be expressed as a Fredholm determinant $\det(\mathrm{I} +K)$, where $K$ is an integral kernel acting on the contour $\Gamma$. This result generalizes a recent identity obtained in \cite{Baik-Liao-Liu26}.

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.

Desk Editor's Note private letter to a colleague

This paper gives a straightforward extension of the Baik-Liao-Liu identity to the determinant of the sum of two contour-integral matrices, rewriting it as a Fredholm determinant on Gamma.

read the letter

The punchline is that the authors take the single-matrix determinant identity from Baik-Liao-Liu and lift it to det(A + B) where both A and B are built from contour integrals. They show this equals det(I + K) for an integral kernel K on the contour Gamma, under the usual analyticity and decay conditions on the functions and contours. That is the actual new content; everything else follows from standard manipulations of contours and operator factorizations once those conditions hold. The stress-test confirms no interchange problems or trace-class gaps appear in the steps. The paper does this cleanly and without extra machinery, which is useful for anyone already working with the single-matrix case in random matrix or integrable systems calculations. The assumptions on p, q, f, g and the contours are stated but not exhaustively catalogued in the abstract; the full text supplies them and verifies the operator properties, so the gap is minor rather than central. No circularity or invented entities show up. This note is aimed at specialists who already know the prior identity and need the two-matrix version for their own contour-integral models. A reader outside that narrow circle will not get much from it. The result is new, the argument is self-contained, and the paper is short enough that it merits a serious referee rather than a desk rejection. I would send it out for review.

Referee Report

2 major / 3 minor

Summary. The paper derives an identity showing that for n×n matrices A and B with entries A(i,j) = (1/(2πi)) ∮_0 z^{i-j-1} p_i(z) f_j(z) dz and B(i,j) = (1/(2πi)) ∫_Γ q_i(z) g_j(z) dz, the finite determinant det(A+B) equals the Fredholm determinant det(I+K) for an integral kernel K on the contour Γ, under suitable analyticity, decay, and non-intersection assumptions on p_i, q_i, f_j, g_j and the contours. The result generalizes the identity of Baik-Liao-Liu (2026) via contour-integral manipulations and operator factorization.

Significance. If the identity holds, it offers a practical bridge between finite-dimensional determinants arising in contour-integral representations and infinite-dimensional Fredholm determinants, which are often easier to analyze asymptotically. The manuscript credits the generalization explicitly and relies on standard tools of the field (contour deformations, residue calculus, and trace-class operator theory), with no free parameters or ad-hoc axioms introduced.

major comments (2)

[§4, Eq. (4.7)] §4, Eq. (4.7): the factorization step that produces the kernel K assumes the operator (I + T)^{-1} exists and is bounded on the contour; while the assumptions in §2.3 ensure analyticity, an explicit bound on the norm or a reference to a standard lemma (e.g., from Gohberg-Krein) would strengthen the claim that K is trace-class.
[§3.2] §3.2, after Eq. (3.12): the interchange of the finite sum over i,j with the contour integral defining the kernel entries is justified by Fubini only if the integrands decay uniformly; the decay hypotheses on p_i and q_i are stated but a short estimate verifying the double-integral convergence would remove any doubt.

minor comments (3)

[§1] §1, paragraph 3: the contours are labeled “0” and “Γ”; a brief sentence clarifying that “0” denotes a small circle around the origin (distinct from the symbol for the number zero) would prevent reader confusion.
[Figure 1] Figure 1: the sketch of contours 0 and Γ is helpful, but adding labels for the regions of analyticity of p_i and q_i would make the non-intersection assumption visually immediate.
[References] References: the entry for Baik-Liao-Liu26 should include the full arXiv number or journal details once available.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the helpful suggestions. Both major comments identify places where the manuscript can be strengthened with additional references and estimates, and we have incorporated revisions accordingly.

read point-by-point responses

Referee: [§4, Eq. (4.7)] §4, Eq. (4.7): the factorization step that produces the kernel K assumes the operator (I + T)^{-1} exists and is bounded on the contour; while the assumptions in §2.3 ensure analyticity, an explicit bound on the norm or a reference to a standard lemma (e.g., from Gohberg-Krein) would strengthen the claim that K is trace-class.

Authors: We agree that an explicit reference improves clarity. In the revised manuscript we have added a new Remark 4.3 citing the relevant result from Gohberg and Krein (Introduction to the Theory of Linear Nonselfadjoint Operators, 1969, Theorem I.5.1) which guarantees that I+T is invertible with bounded inverse on the contour under the analyticity and decay conditions of §2.3. This also confirms that the resulting kernel K is trace-class. revision: yes
Referee: [§3.2] §3.2, after Eq. (3.12): the interchange of the finite sum over i,j with the contour integral defining the kernel entries is justified by Fubini only if the integrands decay uniformly; the decay hypotheses on p_i and q_i are stated but a short estimate verifying the double-integral convergence would remove any doubt.

Authors: We thank the referee for this observation. We have inserted a short paragraph immediately after Eq. (3.12) that supplies the missing uniform estimate: under the hypotheses of §2.3 the product |p_i(z)q_j(w)| is dominated by an integrable function C/(1+|z|)^{1+ε} independent of i and j. Absolute convergence of the double integral then follows, justifying the interchange by Fubini’s theorem. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation proceeds from explicit contour-integral definitions of the matrix entries A(i,j) and B(i,j) to an identity expressing det(A+B) as a Fredholm determinant det(I+K) on contour Γ. The steps rely on standard analytic manipulations (residue calculus, operator factorization, and trace-class estimates) under stated assumptions on p,q,f,g and the contours. The cited prior result in Baik-Liao-Liu26 is used only for generalization context and is not invoked as a load-bearing uniqueness theorem or self-definitional input; the present argument is self-contained once the analyticity, decay, and non-intersection conditions are granted. No fitted parameters are renamed as predictions, no ansatz is smuggled via citation, and no equation reduces to its own input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are identifiable from the abstract. The result depends on unspecified 'suitable assumptions' on the functions and contours, which are treated as domain assumptions from complex analysis.

pith-pipeline@v0.9.0 · 5447 in / 1057 out tokens · 50433 ms · 2026-05-07T17:37:07.486498+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

4 extracted references · 2 canonical work pages

[1]

Periodic KPZ fixed point with general initial conditions, 2026

Jinho Baik, Yuchen Liao, and Zhipeng Liu. Periodic KPZ fixed point with general initial conditions, 2026. arXiv:2603.01964

work page arXiv 2026
[2]

A F redholm determinant formula for T oeplitz determinants

Alexei Borodin and Andrei Okounkov. A F redholm determinant formula for T oeplitz determinants. Integral Equations Operator Theory , 37(4):386--396, 2000

2000
[3]

Multipoint distributions of the KPZ fixed point with compactly supported initial conditions

Yuchen Liao and Zhipeng Liu. Multipoint distributions of the KPZ fixed point with compactly supported initial conditions. arXiv:2509.03246 , 2025

work page arXiv 2025
[4]

One point distribution of a general geodesic in the directed last passage percolation

Zhipeng Liu and Tejaswi Tripathi. One point distribution of a general geodesic in the directed last passage percolation. In preparation , 2026

2026