Moments of Random Permanents and Hafnians: A Replica Approach

Angelo Mariano; Antonello Scardicchio; Giorgio Parisi; Paolo Facchi; Saverio Pascazio

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Moments of Random Permanents and Hafnians: A Replica Approach

Paolo Facchi, Angelo Mariano, Giorgio Parisi, Saverio Pascazio, Antonello Scardicchio

Disordered Systems and Neural Networks · Submitted 20 Oct 2026

Pith's verdict: Closed-form moment asymptotics for random permanents and Hafnians via replicas, with solid numerics and one honest proof gap: the dominant complex saddle is only checked for k≤5.

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arXiv ID2610.27872 (version 2)

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CitationAngelo Mariano, Antonello Scardicchio, Giorgio Parisi, Paolo Facchi, Saverio Pascazio. "Moments of Random Permanents and Hafnians: A Replica Approach." arXiv:2610.27872 (2026). https://pith.science/paper/Z4XNCLAR

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Punchline: worth a serious look, and the reader's conditional verdict is the right one. This paper derives closed-form asymptotics — exponential rate, prefactor, and first 1/N correction — for the 2k-th moments of random permanents and Hafnians via a Hubbard-Stratonovich replica treatment. The central identities (22), (43), and (59) are exact, the saddle-point expansion in the Appendix is explicit, and the formulas agree with exact enumerations and exact finite-N values, including the k=2 exact result ⟨Y²⟩=N+1 and the 1/N coefficient fits. That is real evidence, not numerology.

What is new: the replica derivation itself, the real-Gaussian and Rademacher asymptotics, the Hafnian results, the 1/N corrections, and the k=2 transition with its power-law tail. The complex-permanent rate and the moment-size duality were already in Nezami [39]; the paper cites him and uses his results for comparison. The dominant-saddle step in the real case is supported by a cited published result [38]; that part is fine.

Two soft spots, in proportion. First, the complex-case asymptotics rest on the claim that the replica-symmetric point y_ab=y0 is the global maximum of |per(y)| on the sphere Σ|y|²=k. In Section IV.A the paper says it checked this numerically for k=3,4,5 and that it should follow from a Hadamard-type inequality in [40], but no derivation is supplied. For arbitrary k the premise is unproved, and Eqs. (49)–(52) plus the Hafnian rates via (61) inherit it. This is a proof gap rather than an observed inconsistency; the exact identities and tables make the statement plausible. The paper's own sentence flags the missing support, and a conditional verdict is the honest response.

Second, the Appendix discloses that much of the algebra was generated by Claude and not hand-verified. That disclosure is candid, and the cross-checks are real: propagators against direct Hessian inversion, Wick contractions against brute-force tensor sums, and final coefficients against exact moments. But “checked by an adversarial LLM” is not a mathematical check; the most delicate sums deserve hand verification before the formulas are treated as rigorous.

The photonics section is properly scoped — lossless, collision-free, many-mode assumptions are stated, and the authors explicitly leave the bulk distribution and anti-concentration alone. Audience: random matrix theory and boson-sampling people. It deserves a serious referee. My recommendation: send it to review, ask for a proof (or a clearly stated conjecture) of the complex-saddle global-maximum step, and request independent verification of the machine-generated pieces.

Referee report

Significance

If the claimed asymptotic formulas hold, this is a substantial advance: it provides closed-form leading asymptotics for random permanents and Hafnians, including prefactors and 1/N corrections, with no fitted parameters. The exact identities (22), (24), (43), and (59) are proven in the text, and the numerical agreement with exact enumerations in Tables I-IV is strong evidence that the saddle-point expansion is correct. The moment-size duality (44) and the Hafnian-permanent duality (59) are elegant and useful, and the derivation is considerably simpler than the representation-theoretic route in [39]. The authors also deserve credit for disclosing the machine-assisted algebraic calculations in the Appendix and for stating the limitations of the bulk-distribution and anti-concentration results. The main obstacle to full rigor is the unproved global-extremum premise for |per(y)| on the sphere for general k in the complex case, which affects the central asymptotic claim (50) and, through (62)-(63), the Hafnian rates.

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The paper

The paper claims

The 2k-th moments of permanents and Hafnians of random matrices are computed by a replica-symmetric saddle point for k>2, including prefactors and first 1/N corrections.

The reading

A permanent is like a determinant without minus signs, and a Hafnian is its pairing counterpart. Both count products of matrix entries over permutations or pairings, and both appear in the probabilities of bosonic photon experiments. For a random matrix with independent entries, the moments of the permanent describe the shape of its output distribution, but high moments were previously difficult to access.

The authors write the permanent as a contour integral, average over the random matrix, and then use the replica trick from spin-glass theory. The 2k copies of the permanent interact through Hubbard-Stratonovich fields, and for large matrix size N the integral is controlled by a single replica-symmetric saddle point. This produces closed-form formulas for the exponential growth rate, the polynomial prefactor, and the first 1/N correction for real and complex entries. The growth rates are universal: they depend only on whether the entries are real or complex, while higher cumulants such as kurtosis only change the prefactor and subleading terms.

The same machinery gives dualities connecting permanents of complex Gaussian matrices, permanents of real Gaussian matrices, and Hafnians of symmetric complex matrices. Applied to boson sampling, the formulas estimate how many samples are needed to measure cross-entropy scores, and they predict a power-law tail of the output-probability distribution near the bulk. The paper does not compute the bulk distribution itself, which requires replica symmetry breaking, and the photonics conclusions assume lossless collision-free setups and Haar-random interferometers with many modes.

Extended reading notes

Core claim

Central load-bearing claim: for k>2 the 2k-th moment of the permanent of an N x N matrix with independent entries is controlled by a replica-symmetric saddle point and has closed-form asymptotics, for real entries <Y^k> = C_R^k (1 + a_R^1/N + O(N^-2)) N^{k/2} e^{N Lambda_R(k)} (Eq. 30) and for complex entries a similar expression with Lambda_C(k) (Eq. 50). If true, this provides exact leading asymptotic theory for random permanents and Hafnians, including universal rates depending only on real vs complex entries, and the k=2 transition separating a power-law tail from the bulk.

Load-bearing premise

The saddle-point evaluation of the complex permanent integral (46) assumes the replica-symmetric point y_ab = y0 is the unique dominant critical point, i.e. |per(y)| is globally maximized on the sphere sum |y_ab|^2 = k at y_ab = y0. The paper reports only numerical checks for k=3,4,5 and says the statement should follow from a Hadamard-type inequality in [40]; for arbitrary k this global-extremum premise is unproved, and the rate and prefactor formulas for the complex ensemble and for Hafnians inherit it.

Original abstract

Using contour-integral representations of the permanent and of the Hafnian, we turn the computation of their moments for random matrices into a replica calculation, analogous to that of the moments of the partition function of a mean-field spin glass. For the $2k$-th moment of $N\times N$ matrices with independent entries, the dominant saddle point is replica symmetric for $k>2$, and we obtain in closed form the exponential growth rate, the prefactor and the first $1/N$ correction, in agreement with exact enumerations. The growth rate depends only on whether the entries are real or complex; higher cumulants affect only the prefactor and the $1/N$ term. At $k=2$ the replica-symmetric solution becomes unstable, separating a power-law bulk of the distribution from a large-deviation tail, which we compute. The results apply to boson sampling with single photons and with squeezed light.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-10-21: 1 request, 4,386 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.48 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.02 · states quantitative result 0.66 · compares with prior results 0.82 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

Using contour-integral representations of the permanent and of the Hafnian, we turn the computation of their moments for random matrices into a replica calculation, analogous to that of the moments of the partition function of a mean-field spin glass. For the $2k$-th moment of $N\times N$ matrices with independent entries, the dominant saddle point is replica symmetric for $k>2$, and we obtain in closed form the exponential growth rate, the prefactor and the first $1/N$ correction, in agreement with exact enumerations. The growth rate depends only on whether the entries are real or complex; higher cumulants affect only the prefactor and the $1/N$ term. At $k=2$ the replica-symmetric solution becomes unstable, separating a power-law bulk of the distribution from a large-deviation tail, which we compute. The results apply to boson sampling with single photons and with squeezed light.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Using contour-integral representations of the permanent and of the Hafnian, we turn the computation of their moments for random matrices into a replica calculation, analogous to that of the moments of the partition function of a mean-field spin glass.	method 0.92	0.93	0.13	asserted 1.00
s1	For the $2k$-th moment of $N\times N$ matrices with independent entries, the dominant saddle point is replica symmetric for $k>2$, and we obtain in closed form the exponential growth rate, the prefactor and the first $1/N$ correction, in agreement with exact enumerations.	result 0.99	0.94	0.42	asserted 1.00
s2	The growth rate depends only on whether the entries are real or complex; higher cumulants affect only the prefactor and the $1/N$ term.	result 0.91	0.89	0.17	asserted 1.00
s3	At $k=2$ the replica-symmetric solution becomes unstable, separating a power-law bulk of the distribution from a large-deviation tail, which we compute.	result 1.00	0.94	0.14	asserted 1.00
s4	The results apply to boson sampling with single photons and with squeezed light.	interpretation 0.93	0.48	0.08	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** FIG. 1. Scheme of a boson sampling experiment. Photons are

**Figure 3.** FIG. 3. The full counting of the

**Figure 5.** FIG. 5. Identical single-mode squeezed vacuum states are

**Figure 6.** FIG. 6. Diagrams of order 1

Showing 5 of the paper's 6 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:2610.27872 by the authors.

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