{"id":"6bfb7d5c-7eb9-47ee-a34a-edb476abe790","arxiv_id":"1908.09408","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A new spherical transform with sign parameters gives the joint eigenvalue density and kernels for products of Pólya ensembles with Hermitian matrices.","lead":"This paper derives exact formulas for the eigenvalue statistics of products of random rectangular matrices with Hermitian matrices, covering both fixed and random Hermitian factors. It does so by building a new harmonic-analysis toolkit that avoids difficult group integrals.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Appendix A2's factorization proof requires a QR decomposition k'g = t Π_{l,m} \\tilde k with t invertible triangular, impossible when n1<min(l,m); Eq. (27) and thus Theorems IV.5/IV.8 lack proof in the advertised rank-deficient regime.","rationale":"Proposition III.4 is the keystone: Corollary III.8 converts the multiplicative convolution to products of spherical transforms, and Theorem IV.5 and Proposition IV.8 invert that transform to obtain explicit JPDFs and kernels. If the factorization constant or shifted parameters in Eq. (27) are wrong for rank-deficient g, every subsequent formula restricted to that regime fails. The proof in Appendix A2 is the least secure point: it silently assumes an LQ-type decomposition for every k'g with t invertible, which is impossible when the rank is below min(l,m). The explicit example computed shows the formula is plausible and the obstruction can be of measure zero, so the paper is not necessarily incorrect; however the proof as written does not establish the advertised generalization, and the reader's conditional verdict is the right one. No independent machine-checked verification is provided, and the cited prior work [19] covers the rectangular G-convolution but not the H-convolution in Eq. (27). Thus the central claim is likely correct but currently rests on an unproven decomposition; a revision supplying the almost-everywhere decomposition lemma, or a direct SVD-based proof of Eq. (27), would settle it.","tokens_in":30720,"tokens_out":23134,"duration_ms":225960,"concrete_test":"Independently re-derive Eq. (27) for a rank-deficient case, e.g. l=m=2, n1=1, n2=2, using the singular-value decomposition g = u diag(σ,0) v^* and explicit integration over U(2), without invoking the QR step of Appendix A2. If the constant and shifted arguments \\tilde s, \\tilde L obtained differ from C_{m,r+|n1-n2|}(\\tilde s) in Eq. (27), Proposition III.4 is false; if they match, verify the same for l=3,m=2,n1=1 by Monte Carlo integration, confirming that the gap is only a missing statement that the leading r×r principal minor of k'g is nonzero almost everywhere.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central new tool is the factorization in Proposition III.4. In Appendix A2 (Eqs. A9-A10) the proof performs a 'QR-decomposition' k'g = t Π_{l,m} \\tilde k with t ∈ T_l, where T_l is defined in Section II as the group of invertible lower triangular matrices. But t Π_{l,m} has rank min(l,m) for every k'∈K_l and g∈G^{(n1)}_{l,m}, whereas k'g has rank n1. Hence when n1 < min(l,m) - the fixed-rank, genuinely rectangular case, including the projections and inclusions of Example IV.9 - no such decomposition exists, and the derivation of the shifted spherical parameters \\tilde s, \\tilde L and the constant C_{m,r+|n1-n2|}(\\tilde s) in Eq. (27) is unjustified. A direct rank-1 check (l=m=2, n1=1, n2=2, g=e_2e_1^*, x=diag(1,-1)) shows Eq. (27) itself can still hold (both sides equal 1/3 at s=2, L=0), so the issue is a missing almost-everywhere decomposition lemma rather than an evident counterexample. But as written, the central JPDF claims in Theorem IV.5 and Proposition IV.8 rest on an unproven step in exactly the rank-deficient setting the paper advertises.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a harmonic-analysis framework for products of complex rectangular random matrices with Hermitian matrices of fixed rank. The main new ingredient is a spherical transform on fixed-rank Hermitian matrices, defined with an additional sign parameter L in Definition III.1 and inverted in Proposition III.9, together with a factorization formula for the corresponding spherical functions in Proposition III.4. Using this machinery, the author derives explicit joint eigenvalue densities for a Pólya ensemble on rectangular complex matrices multiplied by a fixed Hermitian matrix (Theorem IV.5) and by a polynomial ensemble on Hermitian matrices (Theorem IV.7), and gives transformation formulas for the biorthogonal functions and kernels (Proposition IV.8). The claimed scope includes genuinely rectangular and rank-deficient matrices, and the paper emphasizes that the approach avoids non-compact group integrals and unifies several earlier results.","tokens_in":30989,"tokens_out":9065,"duration_ms":88900,"significance":"If the central claims hold, this is a valuable and nontrivial extension of the spherical-transform approach from products of square complex matrices to products involving fixed-rank Hermitian matrices and rectangular complex matrices. The explicit finite-dimensional formulas for the joint densities and biorthogonal kernels cover a broad class of Pólya ensembles, including Jacobi, Cauchy-Lorentz, and Muttalib-Borodin weights, and they generalize the Hermite-type product ensembles of Forrester, Ipsen, and Liu. The paper also contains detailed, largely self-contained appendix proofs of the explicit spherical functions and the inversion theorem, and it clearly identifies the two-copy complex-plane structure that the sign parameter L encodes. The main advertised tool — the factorization in Proposition III.4 — is exactly what makes the later convolution results possible, so its correctness is load-bearing for the paper's central claims.","major_comments":[{"comment":"The proof of the factorization formula assumes that every l×m matrix k'g with g ∈ G^{(n1)}_{l,m} admits a QR-type decomposition k'g = t Π_{l,m} \\tilde k with t ∈ T_l invertible lower triangular and \\tilde k ∈ K_m. When n1 < min(l,m), this decomposition cannot exist: k'g has rank n1, while t Π_{l,m} has rank min(l,m) for every invertible t. Since Section II explicitly advertises the lower-rank case and Proposition III.4 states no additional restriction, the derivation of the shifted parameters (\\tilde s, \\tilde L) and of the constant C_{m,r+|n1-n2|}(\\tilde s) in Eq. (27) is unjustified precisely in the rank-deficient regime advertised by the paper. Corollary III.8, Theorem IV.5, Theorem IV.7, and Proposition IV.8 inherit this gap. The identity itself may well be true almost everywhere, as a direct rank-1 test suggests, but the manuscript needs either a correct almost-everywhere decomposition lemma for rank-n1 matrices (with a rectangular or non-invertible triangular factor) or an independent proof of Eq. (27) in the rank-deficient case.","section":"Appendix A2, Proposition III.4, Eqs. (A9)-(A10), (27)"}],"minor_comments":[{"comment":"The stated condition \"l, m ≤ n1\" is inconsistent with the definition of G^{(n1)}_{l,m} in Section II, which requires n1 ≤ l,m. As printed, the theorem has no admissible matrices; the intended condition is presumably n1 ≤ l,m and n2 ≤ m. Please correct this and state the exact domain of all four parameters.","section":"Theorem IV.5"},{"comment":"The definition of \\tilde L contains a parenthesis error: \"diag ( L + |n1 − n2)11r, L(|n1−n2|))\" should read \"diag( L + |n1−n2| 11_r, L^{(|n1−n2|)} )\" or an analogous expression.","section":"Proposition III.4, Eq. (27)"},{"comment":"The spherical function Ψ is defined in Definition III.1 without an L argument, but Eq. (28) writes Ψ(s,L; ·). Either define the L-dependent version or remove the second argument for consistency.","section":"Proposition III.4, Eq. (28)"},{"comment":"In the second line of Eq. (37), the notation \"SΨ PG(\\tilde s) SΦ PH(s,L)\" should refer to QH, not PH, in the second factor; as written, the Hermitian input is incorrectly labeled.","section":"Corollary III.8, Eq. (37)"},{"comment":"The displayed formula for the integral over C^{l×m} has a missing factor or a typographical issue in the relation between d\\tilde g and the measure on G^{(m)}_{l,m}; please check the constant in Eq. (5), since the equivalence of measures is used later in the paper.","section":"Section II, Remark II.1"}],"recommendation":"major_revision","confidential_remarks":"The paper's central framework is promising and the stress-test example indicates that the disputed factorization may be true in the rank-deficient case, so this is not a clear-cut rejection. However, the proof gap in Proposition III.4 is exactly at the point that supports the paper's advertised scope. I would be willing to look at a revised version that supplies a correct decomposition lemma or an alternative proof for n1 < min(l,m). I also noticed a few minor typographical issues in the parameter definitions and theorem statements that should be corrected."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First thing you should know: this paper is worth engaging with, but it has a real proof gap in the key factorization result for exactly the rank-deficient case it advertises. The new object is the spherical function on fixed-rank Hermitian matrices with an extra binary label L for eigenvalue signs, and the paper uses it to get finite-size JPDFs, biorthogonal functions, and kernels for products of Pólya ensembles with fixed or random Hermitian matrices. That genuinely goes beyond the earlier GUE case and gives a unified framework. The explicit formulas in Theorem IV.5 and Proposition IV.8 look right, and the reduction to the known GUE/induced-Ginibre results is a good sanity check.\n\nThe soft spot is in Appendix A2, proof of Proposition III.4. The proof assumes a 'QR-decomposition' k'g = t Π_{l,m} \\tilde k with t ∈ T_l invertible lower triangular. For g of rank n1 < min(l,m), the left side has rank n1, while t Π_{l,m} \\tilde k has rank min(l,m). So no such decomposition exists. This is not cosmetic: the factorization formula (27) feeds Corollary III.8 and then Theorem IV.5 and Proposition IV.8. The stress-test rank-1 example (l=m=2, n1=1, n2=2) shows the formula itself can still hold — both sides evaluate to 1/3 at s=2, L=0 — so I do not think the claim is false. It looks like a missing limiting argument: the rank-deficient case should follow from the full-rank case via the rank-reduction relations in Remark III.2, but that argument is not in the paper.\n\nOther concerns are minor. The spherical transform for rectangular Pólya ensembles is cited from earlier work rather than proven here; for a paper whose purpose is to extend to rectangular matrices, that is acceptable if the citation really covers it, but a precise pointer would help. The conditions in Theorem IV.5 as printed ('l, m ≤ n1') cannot be right if G^(n1)_{l,m} is to be nonempty, and the normalization constants deserve a careful check.\n\nBottom line: this is a serious paper by someone who knows the area, and the main results are likely correct. But the proof of the central factorization is incomplete in the rank-deficient regime. I would send it to a knowledgeable referee and ask for a revision that supplies the missing decomposition or a proper limiting argument, and clarifies the rectangular transform citation. If those are fixed, it becomes a reliable reference.","headline":"A serious and useful extension of the harmonic-analysis approach to products with Hermitian matrices, but the key factorization proof has a genuine gap in the rank-deficient case it advertises.","tokens_in":31523,"tokens_out":5248,"would_cite":true,"duration_ms":51609,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["15B52","42C05"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper derives exact eigenvalue statistics for products of complex rectangular and Hermitian random matrices by extending the spherical-transform formalism to fixed-rank Hermitian matrices.","keywords":["products of independent random matrices","polynomial ensemble","multiplicative convolution","Pólya frequency functions","spherical transform","bi-orthogonal ensembles","Hermitian matrices","fixed rank"],"falsifier":"Evaluate both sides of Eq. (27) numerically in a small rank-deficient case with $n_1<m$, for example $l=m=2$, $n_1=1$, $n_2=2$, using generic $g$ and Hermitian $x$ with distinct eigenvalues, and check whether the $K_m$-average of $\\Phi(s,L;gkxk^*g^*)$ equals the claimed constant times the product of spherical functions; any discrepancy would show the decomposition hypothesis fails on a set of positive measure.","tokens_in":30489,"feed_emoji":"🧮","tokens_out":8544,"duration_ms":75333,"temperature":0.7,"pith_summary":"This paper extends the harmonic-analysis approach to multiplicative convolution to products in which one factor is a complex rectangular matrix and the other is a Hermitian matrix of fixed rank, not necessarily positive definite. The central move is a spherical transform on fixed-rank Hermitian matrices whose frequency variables carry an extra binary sign label, so the transform records how many eigenvalues are positive and how many are negative. With a factorization formula and an inversion theorem for this transform, the author derives exact joint eigenvalue densities, bi-orthogonal functions, and kernels for such products when the rectangular factor is a Pólya ensemble and the Hermitian factor is fixed or drawn from a polynomial ensemble. The formulas hold at finite matrix dimension and cover both regimes, $n_1\\ge n_2$ and $n_1<n_2$, including the case where the product drops rank.","feed_headline":"Exact eigenvalue formulas for products with Hermitian matrices","feed_subtitle":"A sign-aware spherical transform yields densities and kernels for Pólya and polynomial ensembles at finite size.","key_machinery":"The central object is the spherical function on $H_l^{(n)}$, the $l\\times l$ Hermitian matrices of rank $n$, defined in Eq. (17) as a Haar integral over the unitary group of products of powers of nested determinants $|\\det\\Pi_{j,l}kxk^*\\Pi_{l,j}|^{s_j-s_{j+1}-1}$ times sign factors $[\\operatorname{sign}\\det\\Pi_{j,l}kxk^*\\Pi_{l,j}]^{L_j-L_{j+1}-1}$. The binary labels $L_j\\in\\{0,1\\}$ are what let the transform invert on the whole real line, because they keep track of the signs of the nested determinants. Three identities carry the argument: the explicit determinantal evaluation of $\\Phi$ (Theorem III.3), the factorization of the $K_m$-average of $\\Phi(s,L;gkxk^*g^*)$ into a product of spherical functions times the constant $C_{m,r+|n_1-n_2|}(\\tilde s)$ (Proposition III.4), and the inverse transform with the regularizing factor $\\zeta_n$ (Proposition III.9). Together they reduce the multiplicative convolution to a product of spherical transforms, mirroring how the univariate Mellin transform diagonalizes multiplicative convolution.","core_discovery":"The paper establishes that multiplying a Pólya ensemble on complex rectangular matrices by a fixed or random Hermitian matrix of fixed rank produces another polynomial ensemble, and that the full eigenvalue statistics of the product can be written down explicitly. The enabling object is a modified spherical function on the space $H_l^{(n)}$ of rank-$n$ Hermitian matrices, given in Eq. (17), which pairs each Mellin–Fourier variable $s_j$ with a sign label $L_j$; the two copies of the complex plane are necessary because the numbers of positive and of negative eigenvalues are separately conserved by the conjugation $gxg^*$. Proposition III.4 supplies the factorization of this spherical function under the Haar average over the middle unitary group, with the constant $C_{m,r+|n_1-n_2|}(\\tilde s)$ absorbing the rectangular dimensions and the rank mismatch, and Proposition III.9 gives an explicit inverse. Theorem IV.5 and Proposition IV.8 then deliver the joint probability density, the bi-orthogonal functions, and the kernel in closed form; the transformed weights are explicit one-dimensional convolutions of the Pólya weight with a $\\beta$-type density, and the kernels are contour integrals over the Mellin transform of that weight.","pith_inferences":["One immediate extension the paper leaves implicit is the large-$n$ analysis: the contour-integral kernels of Proposition IV.8 are tailored for saddle-point evaluation, so hard-edge scaling limits for products with a shifted GUE are a natural next target (the author announces a companion paper on this).","The sign-label structure suggests that the same spherical-transform construction could be adapted to the remaining classical symmetric spaces, notably odd-dimensional orthogonal and unitary symplectic adjoint actions, which the conclusions explicitly list as open.","Since the factorization constant is the spherical transform of a random projection, the results also provide a ready-made tool for products with an intervening random projection, the rectangular analogue of truncating a unitary factor."],"forward_implications":["For a fixed Hermitian matrix with eigenvalues $a$, multiplying it by a Pólya-ensemble rectangular matrix produces a polynomial ensemble whose weight is a single integral convolution of the Pólya weight with a beta-type density (Theorems IV.5 and IV.7).","When the Hermitian factor is random, the bi-orthogonal functions and kernel of the product are obtained from those of the original ensemble by explicit contour-integral transformations (Proposition IV.8), so the spectral statistics are known in closed form.","Projections and inclusions of Hermitian matrices, realized as blocks of Haar-distributed unitary matrices, fall within the same framework and no longer require the restrictive dimension inequalities of earlier truncation results (Example IV.9).","The GUE-product case treated earlier via non-compact group integrals emerges as a special case with the Mellin transform $\\Gamma(s)$, so the new derivation is fully algebraic and works for any Pólya weight such as Jacobi, Cauchy–Lorentz, or Muttalib–Borodin weights.","All formulas are valid at finite matrix dimension, which makes them directly testable by numerical simulation of the eigenvalue densities for small $n_1,n_2,l,m$."],"supporting_citations":[{"why":"the GUE-product case the present work generalizes, supplying the benchmark that avoids non-compact group integrals","marker":"[14]"},{"why":"the real antisymmetric analogue whose modified spherical function motivates the construction here","marker":"[23]"},{"why":"defines Pólya ensembles and establishes their multiplicative convolution semi-group, which the paper builds on","marker":"[16]"},{"why":"the classical spherical-transform theory that the paper modifies for fixed-rank Hermitian matrices","marker":"[18]"},{"why":"provides the spherical transform and inversion for Pólya ensembles on square matrices, extended here to rectangular and Hermitian settings","marker":"[25]"},{"why":"gives the convolution closure of Pólya ensembles and the polynomial-ensemble transformation formulas for products","marker":"[26]"},{"why":"introduces polynomial ensembles, the class in which the resulting eigenvalue statistics live","marker":"[31]"},{"why":"supplies the co-rank-1 projected eigenvalue distribution used to prove the explicit form of the spherical function","marker":"[8]"}],"fun_headline_variants":["Polynomial ensembles from rectangular-Hermitian products","Closed-form eigenvalues for Hermitian matrix products","Sign-aware spherical transform yields product formulas","Explicit kernels for rectangular-Hermitian products","Rectangular-Hermitian products yield closed-form spectra"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The derivation rests on the factorization in Proposition III.4, which assumes that every matrix $k'g$ with $k'\\in K_l$ and $g\\in G_{l,m}^{(n_1)}$ admits a decomposition $k'g=t\\,\\Pi_{l,m}\\tilde k$ with $t$ lower triangular and the determinants $\\det(t_jt_j^*)$ nonzero, even when the rank $n_1$ is smaller than $m$; if that decomposition fails on a set of positive Haar measure, the constant $C_{m,r+|n_1-n_2|}(\\tilde s)$ and the derived densities would not be justified.","fun_headline_variants_meta":{"raw":{"variants":["Polynomial ensembles from rectangular-Hermitian products","Closed-form eigenvalues for Hermitian matrix products","Sign-aware spherical transform yields product formulas","Explicit kernels for rectangular-Hermitian products","Rectangular-Hermitian products yield closed-form spectra"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000754,"raw_usage":{"total_tokens":3361,"prompt_tokens":958,"completion_tokens":2403,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":574,"completion_tokens_details":{"reasoning_tokens":2334}},"tokens_in":574,"tokens_out":2403,"duration_ms":18144,"temperature":1.0,"reasoning_tokens":2334,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:13:18.547975+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate both sides of Eq. (27) numerically in a small rank-deficient case with $n_1<m$, for example $l=m=2$, $n_1=1$, $n_2=2$, using generic $g$ and Hermitian $x$ with distinct eigenvalues, and check whether the $K_m$-average of $\\Phi(s,L;gkxk^*g^*)$ equals the claimed constant times the product of spherical functions; any discrepancy would show the decomposition hypothesis fails on a set of positive measure.","supporting_citations":[{"cited_title":"7 This theorem is proven in Appendix A 1 in a very similar way as t he real counterpart with even dimensional antisymmetric matrices in [23]","cited_arxiv_id":null,"evidence_quote":"the GUE-product case the present work generalizes, supplying the benchmark that avoids non-compact group integrals"},{"cited_title":"For Gn this statement is equal to the one in [25]","cited_arxiv_id":null,"evidence_quote":"the real antisymmetric analogue whose modified spherical function motivates the construction here"},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"defines Pólya ensembles and establishes their multiplicative convolution semi-group, which the paper builds on"},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"the classical spherical-transform theory that the paper modifies for fixed-rank Hermitian matrices"},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"provides the spherical transform and inversion for Pólya ensembles on square matrices, extended here to rectangular and Hermitian settings"},{"cited_title":", wn ∈ L1 n−1(R) is a K-invariant ensemble whose eigenvalues a ∈ D are distributed as pD(a) = 1 n! ∆ n(a) det[wb(ac)]b,c=1,...,n det[Mwb(c, c − 1)]b,c=1,...,n ∈ L1 Prob(Dn)","cited_arxiv_id":null,"evidence_quote":"gives the convolution closure of Pólya ensembles and the polynomial-ensemble transformation formulas for products"},{"cited_title":", aj−1, aj+1,","cited_arxiv_id":null,"evidence_quote":"introduces polynomial ensembles, the class in which the resulting eigenvalue statistics live"},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"supplies the co-rank-1 projected eigenvalue distribution used to prove the explicit form of the spherical function"}],"review_version":1}