REVIEW 1 major objections 5 minor 86 references
Products of Complex Rectangular and Hermitian Random Matrices
T0 review · 1 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash
Pith's one-line read 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.
desk verdict 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. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
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.
What would settle it
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.
Extended reading notes
Core claim
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.
Load-bearing premise
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.
Editorial extensions
If this is right
- 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 $n_1,n_2,l,m$.
Reading between the lines
- 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.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
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.
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 (1)
- [Appendix A2, Proposition III.4, Eqs. (A9)-(A10), (27)] 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.
minor comments (5)
- [Theorem IV.5] 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.
- [Proposition III.4, Eq. (27)] 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.
- [Proposition III.4, Eq. (28)] 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.
- [Corollary III.8, Eq. (37)] 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 II, Remark II.1] 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.
Circularity Check
No significant circularity: spherical transforms and inverses are proved from definitions; no target result is used as input.
full rationale
The paper's derivation chain is self-contained in the sense relevant to circularity. The spherical functions in Definition III.1 (Eq. 17) are defined as integrals over unitary groups, and their explicit forms (Theorem III.3) and invertibility (Proposition III.9) are proved by recursion, Andréief's identity, and Carlson's theorem rather than assumed. The factorization formula (Proposition III.4, Eq. 27) is derived in Appendix A2 from a QR decomposition and Haar-invariance, with Corollary III.8 following directly. The spherical transforms of Pólya and polynomial ensembles in Proposition IV.2 are either proved in the text or cited to prior published works ([25,26]) whose statements do not include the present convolution results; those citations are independent support and are not restatements of Theorem IV.5 or Proposition IV.8. The JPDFs in Theorem IV.5 follow by applying the factorization to the spherical transform, identifying the resulting Mellin products with the displayed weight functions, and inverting via Proposition III.9; no fitted parameter is renamed as a prediction. Thus no step reduces to its own input by construction. The possible rank-deficient QR gap in Appendix A2 is a correctness concern rather than a circularity, since the target JPDFs are not assumed in the factorization proof.
Assumptions & free parameters
assumptions (5)
- domain assumption Baryshnikov's interlacing theorem (Theorem A.1) for the eigenvalue distribution of a co-rank-1 projection of a Haar-conjugated Hermitian matrix.
- standard math Multivariate Carlson's theorem for analytic continuation of the spherical function from Re(s_j-s_{j+1}) > 1 to the complex plane.
- domain assumption Standard spherical-transform theory on complex general linear groups, including the Gelfand-Naïmark integral and factorization (26), from Helgason [18] and prior work.
- standard math Mellin transform on R with sign label L and its inversion (Eqs. (12)-(13)) for L1 functions.
- domain assumption Spherical transform formulas for Pólya ensembles on rectangular matrices (Prop IV.2(2)), proven for the square case in prior work [25,26].
Cite this review
Pith. "Pith review of Products of Complex Rectangular and Hermitian Random Matrices." pith.science (2026). https://pith.science/paper/TG46PWEU
@misc{pith2026190809408,
author = {Pith},
title = {Pith review of: Products of Complex Rectangular and Hermitian Random Matrices},
year = {2026},
howpublished = {\url{https://pith.science/paper/TG46PWEU}},
note = {Machine review of arXiv:1908.09408}
}
read the original abstract
Products and sums of random matrices have seen a rapid development in the past decade due to various analytical techniques available. Two of these are the harmonic analysis approach and the concept of polynomial ensembles. Very recently, it has been shown for products of real matrices with anti-symmetric matrices of even dimension that the traditional harmonic analysis on matrix groups developed by Harish-Chandra et al. needs to be modified when considering the group action on general symmetric spaces of matrices. In the present work, we consider the product of complex random matrices with Hermitian matrices, in particular the former can be also rectangular while the latter has not to be positive definite and is considered as a fixed matrix as well as a random matrix. This generalises an approach for products involving the Gaussian unitary ensemble (GUE) and circumvents the use there of non-compact group integrals. We derive the joint probability density function of the real eigenvalues and, additionally, prove transformation formulas for the bi-orthogonal functions and kernels.
Reference graph
Works this paper leans on
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[1]
the complex general linear group of dimension n × n: Gn = Gl C(n) equipped with the flat Lebesgue measure dg,
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[2]
the n × n invertible Hermitian matrices Hn equipped with the flat Lebesgue measure dx,
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the n × n real diagonal and invertible n × n matrices: Dn ≃ Rn equipped with the flat Lebesque measure da,
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the n × n positive real diagonal n × n matrices An ≃ Rn + equipped with the flat Lebesque measure da,
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the group of the complex lower n × n triangular matrices Tn = { t ∈ G ⏐ ⏐ ⏐ ⏐t = {tab}a,b=1...,n with tij ∈ C, t ij = 0 for i < j } equipped with the flat Lebesgue measure dt,
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the unitary group Kn = U(n) equipped with the normalized Haar measure d∗k, 3
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the complex l × m matrices of rank n ≤ l, m G(n) l,m = {kΠ l,ngΠ n,m˜k∗|k ∈ Kl, ˜k ∈ Km, and g ∈ Gn}, where G(n) n,n = Gn and equipped with the measure dg′ = d(kΠ l,ngΠ n,m˜k∗) = dgd∗kd∗˜k induced by those on Gn, Kl and Km,
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the l × l Hermitan matrices of rank n ≤ l H (n) l = {kΠ l,nxΠ n,lk∗|k ∈ Kl and x ∈ Hn}, where H (n) n = Hn and equipped with the measure dx′ = d(kΠ l,nxΠ n,lk∗) = dxd∗k induced by those on Hn and Kl,
Show all 86 references
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[9]
and the symmetric group Sn permuting n elements. The matrices of lower rank than their matrix dimensions are o f particular interest since they allow to deal with products of the form g1g2 with g1 ∈ Cl,n, g2 ∈ Cn,m and l, m > n as well as of the form gxg ∗ with g ∈ Cl,n, x ∈ H...
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[10]
, sn) ∈ Cn and L = diag (L1,
the spherical function on H (n) l by Φ( s, L; x) = ∫ Kl d∗k∏ n j=1 sign[det Π j,lkxk∗Π l,j]Lj −Lj+1−1| det Π j,lkxk∗Π l,j|sj −sj+1−1 ∫ Kl d∗k∏ n j=1 | det Π j,lkΠ l,nΠ n,lk∗Π l,j |sj −sj+1−1 (17) for s = diag ( s1, . . . , sn) ∈ Cn and L = diag (L1, . . . , Ln) ∈ Zn 2 with Re ...
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[18, 25] for Gn, Ψ( s; g) = Φ( s, L(n); gg ∗) (18) for all s ∈ Cn and fixed L(n) as in Eq
and the spherical function on G(n) l,m, cf., Ref. [18, 25] for Gn, Ψ( s; g) = Φ( s, L(n); gg ∗) (18) for all s ∈ Cn and fixed L(n) as in Eq. (16). We would like to underline that we employ here a different conv ention of sn+1 compared to the standard literature [ ? ] where its v...
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al ⁄= ak and sl ⁄= sk for all l ⁄= k, and L ∈ Zn 2
Let a ∈ Dn and s ∈ Cn with non-degenerate spectra, i.e. al ⁄= ak and sl ⁄= sk for all l ⁄= k, and L ∈ Zn 2 . The spherical function (17) has the explicit form Φ( s, L; a) = n−1∏ j=0 j! det[[sign(ac)]Lb |ac|sb ]b,c=1,...,n ∆ n(a)∆ n(s) . (23)
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[13]
The spherical function (18) is Ψ( s; a) = n−1∏ j=0 j! det[asb c ]b,c=1,...,n ∆ n(a)∆ n(s)
Let a ∈ An and s ∈ Cn with non-degenerate spectra. The spherical function (18) is Ψ( s; a) = n−1∏ j=0 j! det[asb c ]b,c=1,...,n ∆ n(a)∆ n(s) . (24)
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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]
The normalization (19) is equal to Cl,n(s) = n∏ j=1 (l − j)!Γ[ sj + 1] (n − j)!Γ[ sj + l − n + 1] (25) for any s ∈ Cn. 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]. This theorem sho...
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Additionally, we choose ˜s ∈ Cr and ˜L ∈ {0, 1}r
Let g ∈ G(n1) l,m and x ∈ H (n2) m with r = min {n1, n2} the rank of gxg ∗. Additionally, we choose ˜s ∈ Cr and ˜L ∈ {0, 1}r. Then, we find the following factorization ∫ Km d∗kΦ( s, L; gkxk ∗g∗) = Cm,r+|n1−n2|(˜s) { Ψ( s; g) Φ(˜s, ˜L; x), n 1 = r, Ψ(˜s; g)Φ( s, L; x), n 2 = r (...
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[16]
We defined ˜s = diag ( s + |n1 − n2|11r, s(|n1−n2|)) and ˜L = diag ( L + |n1 − n2)11r, L(|n1−n2|))
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[17]
Let g1 ∈ G(n1) l,m and g2 ∈ G(n2) m,o with r = min{n1, n2} the rank of g1g2. Then, the factorization formula reads ∫ Km d∗kΨ( s, L; g1kg2) = Cm,r+|n1−n2|(˜s) { Ψ( s; g1)Ψ(˜s; g2), n 1 = r, Ψ(˜s; g1)Ψ( s; g2), n 2 = r (28) for all s ∈ Cr, where ˜s is defined as before. The secon...
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[18]
The spherical transform SΦ : L1,K(H (n) m ) → S Φ (L1,K(H (n) m )) corresponding to Φ is defined as SΦ PH (s, L) = ∫ H(n) m dxPH (x)Φ( s, L; x) = n−1∏ j=0 j! ∫ Dn da pD(a) det[[sign(ac)]Lb |ac|sb ]b,c=1,...,n ∆ n(a)∆ n(s) = SΦ pD(s, L) (32) for those s ∈ Cn for which th...
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In the second equalities, we slightly abuse notation and hav e to assume that sl ⁄= sk for l ⁄= k
The spherical transform SΨ : L1,K(G(n) l,m) → S Ψ (L1,K (G(n) l,,m)) corresponding to Ψ is (see [18, 25] for Gn) SΨ QG(s) = ∫ G(n) l,m dgQG(g)Ψ n(s; g) = n−1∏ j=0 j! ∫ An da qA(a) det[asb c ]b,c=1,...,n ∆ n(a)∆ n(s) = SΨ qA(s) (33) for any s ∈ Cn where the integral exi...
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[20]
Let PG ∈ L1,K(G(n1) l,m ) and QH ∈ L1,K(H (n2) m ) with r = min {n1, n2}. Then the spherical transform of the convolution PG ⊛ QH is SΦ [PG ⊛ QH ](s, L) = Cm,r+|n1−n2|(˜s) { SΨ PG(s)SΦ QH(˜s, ˜L), n 1 = r, SΨ PG(˜s)SΦ PH (s, L), n 2 = r, (37) with ˜s = diag (s + |n1 − n2|11r, ...
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For PG ∈ L1,K(G(n1) l,m ) and QG ∈ L1,K(G(n2) m,o ) with r = min {n1, n2}, the spherical transform of the convolution PG ⊛ QG becomes SΨ [PG ⊛ QG](s) = Cm,r+|n1−n2|(˜s) { SΨ PG(s)SΨ QG(˜s), n 1 = r, SΨ PG(˜s)SΨ QG(s), n 2 = r, (38) where ˜s = diag (s + |n1 − n2|11r, s(|n1−n2|)...
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The spherical transform SΦ is injective and, hence, invertible when restricted to its image
Let PH ∈ L1,K(H (n) m ) and pD = IH PH ∈ L1,S(Dn). The spherical transform SΦ is injective and, hence, invertible when restricted to its image. The inverse has the exp licit form pD(a) = S−1 Φ [SΦ pD](a) = ∑ L∈{0,1}n ∆ n(a) (n!)2∏ n−1 j=0 j! lim ǫ→0 ∫ Rn ds (4π)n SΦ pD(ıs + s(...
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For Gn this statement is equal to the one in [25]
Choosing PG ∈ L1,K(G(n) l,m) and pA = IGPH ∈ L1,S(An), the spherical transform SΨ is invertible when restricted to its image with the inverse pA(a) = S−1 Ψ [SΨ pA](a) = ∆ n(a) (n!)2∏ n−1 j=0 j! lim ǫ→0 ∫ Rn ds (2π)n SΨ pA(ıs + s(n)) n∏ l=1 ζn(ǫsl) × ∆ n(ıs + s(n)) det[a−ısb−n+...
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, wn ∈ L1 n−1(R+) is a K-invariant ensemble whose squared singular values a ∈ A are distributed as pA(a) = 1 n! ∆ n(a) det[wb(ac)]b,c=1,...,n det[Mwb(c)]b,c=1,...,n ∈ L1 Prob(An)
A polynomial ensemble on G(n) l,m associated to the weights w1, . . . , wn ∈ L1 n−1(R+) is a K-invariant ensemble whose squared singular values a ∈ A are distributed as pA(a) = 1 n! ∆ n(a) det[wb(ac)]b,c=1,...,n det[Mwb(c)]b,c=1,...,n ∈ L1 Prob(An). (47)
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A Pólya ensemble on G(n) l,m associated to the weight ω ∈ P 1 n−1 is a polynomial ensemble with wb(ac) = ( −ac∂c)b−1ω(ac), (48) i.e., pA(a) = 1 ∏ n j=1 j!Mω(j) ∆ n(a) det[(−ac∂c)b−1ω(ac)]b,c=1,...,n. (49)
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, 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)
A polynomial ensemble on H (n) m associated to the weights w1, . . . , 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). (50) The true potential of the ens...
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, wn ∈ L1 n−1(R+)
Let PG the distribution of a polynomial ensemble on G(n) l,m associated to the weights w1, . . . , wn ∈ L1 n−1(R+). Then, its spherical transform is (see [25, 26] for the square ca se) SΨ PG(s) = n−1∏ j=0 j! det[Mwb(sc + 1)]b,c=1,...,n ∆ n(s) det[Mwb(c)]b,c=1,...,n , (...
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The spherical transform of the distribution PG, which is a Pólya ensemble on G(n) l,m associated to the weight ω ∈ P 1 n−1, is explicitly given by (see [25, 26] for l = m = n) SΨ PG(s) = n∏ j=1 Mω(sj + 1) Mω(n − j + 1) . (52)
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The spherical transform of PH describing a polynomial ensemble on H (n) m associated to the weights w1, . . . , wn ∈ L1 n−1(R) is equal to SΦ PH (s, L) = n−1∏ j=0 j! det[Mwb(sc + 1, Lc)]b,c=1,...,n ∆ n(s) det[Mω(c, c − 1)]b,c=1,...,n (53) with M being the univariate Me...
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(55) and Θ the Heaviside step function
n1 ≥ n2: Then the eigenvalues ˜a of the product gag ∗ are distributed by p(˜a|a) = 1 n2! n2∏ j=1 (m − j)! (m − n1)!(n1 − j)!Mω(n1 − j + 1) ∆ n2(˜a) ∆ n2(a) det [˜ω≥(˜ab|ac)]b,c=1,...,n2 (59) with the weight ˜ω≥(˜ab|ac) = 1 |ac| ˜ω≥ ( ˜ab ac ⏐ ⏐ ⏐ ⏐1 ) = Θ(˜abac) ( ˜ab ...
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, aj−1, aj+1,
n1 < n 2: In this case the joint probability distribution of the eigenv alues ˜a of the product gag ∗ is p(˜a|a) = 1 n1! n1∏ j=1 (m − j)! (m − n1)!(n1 − j)!Mω(n1 − j + 1) ∆ n1 (˜a) ∆ n2 (a) det [ ab−1 c ˜ω<(˜ad|ac) ] b=1,...,n2−n1 d=1,...,n1 c=1,...,n2 = 1 n1! n1∏ ...
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[32]
n1 ≥ n2: The eigenvalues ˜a of the random matrix product gxg ∗ are distributed by p(˜a) = 1 n2! n2∏ j=1 (m − j)! (m − n1)!(n1 − j)!Mω(n1 − j + 1) ∆ n2 (˜a) det [ ∫ ∞ −∞ da˜ω≥(˜ab|a)wc(a) ] b,c=1,...,n2 det [Mwb(c, c − 1)]b,c=1,...,n2 (68) where ˜ω≥ is given in Eq. (60)
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analytical response
n1 < n 2: The joint probability distribution of the eigenvalues ˜a of gag ∗ is p(˜a|a) = 1 n1! n1∏ j=1 (m − j)! (m − n1)!(n1 − j)!Mω(n1 − j + 1) ∆ n1(˜a) det [ Mwc(b, b − 1)∫ ∞ −∞ da˜ω<(˜ad|a)wc(a) ] b=1,...,n2−n1 d=1,...,n1 c=1,...,n2 det [Mwb(c, c − 1)]b,c=1,...,n2 ,...
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n1 ≥ n2: ˜pj(˜a) = ∮ dz 2πiz χ≥(z)pj ( ˜a z ) , ˜qj(˜a) = ∫ ∞ −∞ da |a| ˜ω≥ ( ˜a a ⏐ ⏐ ⏐ ⏐1 ) qj(a), ˜Kn2(˜a1, ˜a2) = ∮ dz 2πiz χ≥(z) ∫ ∞ 0 da a ˜ω≥ ( a| 1) Kn2 ( ˜a1 z , ˜a2 a ) , (73)
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[35]
Mellin-Fourier
n1 < n 2: ˜pj(˜a) =˜an1−n2 ∮ dz 2πiz χ<(z)pn2−n1+j ( ˜a z ) , ˜qj(˜a) = ∫ ∞ −∞ da |a| an2−n1 ˜ω< ( ˜a a ⏐ ⏐ ⏐ ⏐1 ) , ˜Kn1(˜a1, ˜a2) = ( ˜a2 ˜a1 ) n2−n1∮ dz 2πiz χ<(z) ∫ ∞ 0 da an2−n1+1 ˜ω< ( a| 1) Kn2 ( ˜a1 z , ˜a2 a ) . (74) The contour integrals run counter-clockwise around ...
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,0)ˆk∗ ∈ H (n) l with a ∈ Dn and absorb the diagonalizing unitary matrix in the Haar dist ributed matrix k ∈ Kn in Eq
Proof of Theorem III.3 Due to the unitary invariance of the spherical function, we c an diagonalize x = ˆk diag (a, 0, . . . ,0)ˆk∗ ∈ H (n) l with a ∈ Dn and absorb the diagonalizing unitary matrix in the Haar dist ributed matrix k ∈ Kn in Eq. (17). To proceed further, we first...
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Thence, we only need to prove the latter
Proof of Propositions III.4 The factorization formula (28) for Ψ immediately follows from the one for Φ . Thence, we only need to prove the latter. We choose a g ∈ G(n1) l,m and an x ∈ H (n2) m with r = min {n1, n2} the rank of the product g1g2 and consider δsj = sj − sj+1 − 1...
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[38]
Indeed when comparing the second lines of Eqs
Proof of Proposition III.9 Since An is a subset of Dn we can concentrate us on proofing the inverse of the spherical transform of SΦ . Indeed when comparing the second lines of Eqs. (32) and (33) it becom es clear that the two transformations are identical for the domain An sin...
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Proof of Theorem IV.5 Let us denote the distribution of the Pólya ensemble random m atrix g ∈ G(n1) l,m by QG ∈ L1,K Prob(G(n1) l,m ). Then, the distribution PH ∈ L1,K Prob(H (n1) l ) of the product ˜x = gxg ∗ ∈ H (n1) l with x ∈ H (n2) m fixed is certainly PH (˜x|x) = ∫ G(n1) ...
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Comparison of Eq
= a′ 1 n1−n2 ω(0) m−n1(a′ 1). Comparison of Eq. (B4) with the spherical transform (53) of a polynomial ensemble on H (n2) l yields the first part of Theorem IV.5 due to the invertibility of this transform. Let us go over to the opposite case n1 ≤ n2. For this case the spherical...
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Proof of Proposition IV.8 a. The case n1 ≥ n2 The proof of the case n1 ≥ n2 is based on the identity ∫ ∞ −∞ d˜a˜ab ˜ω≥(˜a|a) = ∫ ∞ 0 da′ 1 ∫ ∞ 0 da′ 2ω(n1−n2) m−n1 (a′ 1)a′ 2 n1−n2 ω(a′ 2)(a′ 1a′ 2a)b = (m − n1)!(n1 + b − n2)! (m + b − n2)! Mω(n1 + b − n2 + 1)ab. (B13) Thence,...
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