Pith. sign in

REVIEW 5 major objections 5 minor 36 references

Tensorial free convolution, semicircular, free Poisson and R-transform in high order

T0 review · 5 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Even-order tensors admit a free additive convolution whose R-transform is additive.

desk verdict The tensorial free convolution and Wishart convergence are real contributions, but the proof has a false bound in Proposition 5, the free CLT lacks its freeness hypothesis, and the scope is overstated. read the letter →

arxiv 2412.02572 v2 pith:LO6H3QJV submitted 2024-12-03 math.OA math-phmath.COmath.MPmath.PR

Remi Bonnin This is my paper · ORCID
classification math.OAmath-phmath.COmath.MPmath.PR MSC 46L5415A6960B20
keywords tensorialfreenessfreeconvolutionR-transformsemicircularlawPoissonFuss-CatalannumbersWignertensorWishart
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 aims to build a tensorial analogue of the classical free additive convolution for compactly supported probability measures. The central claim is that if two measures arise from freely independent tensor distributions of even order $p$, their sum defines a measure $\mu \oplus_p \nu$ whose free cumulants are the sums of the individual free cumulants; equivalently, the $R$-transform (the generating function of the free cumulants) satisfies $R_{\mu\oplus_p\nu}(z)=R_\mu(z)+R_\nu(z)$. The operation is commutative and associative, with $\delta_0$ as neutral element. The paper also identifies high-order semicircular and free Poisson laws as the tensor counterparts of the Wigner and Marcenko-Pastur laws, proves Wigner and Wishart tensors converge to them, and gives a free central limit theorem. If correct, this extends free probability's main convolution tool beyond matrices.

What carries the argument

The machinery is the non-crossing poset on $p$-regular trace maps, graphs whose vertices are decorated by tensors and whose edges are contracted according to the tensor indices. The poset order is generated by switches that exchange the endpoints of two edges, with a switch increasing the number of connected components; Moebius inversion on this poset defines the free cumulants $\kappa_b$. The analytic moment-cumulant formula $M_T(z)=C_T(z M_T(z)^{p/2})$ packages moments and cumulants into formal power series, and the additivity of $\oplus_p$ rests on the tensorial freeness criterion for even families: two families are freely independent exactly when all mixed free cumulants vanish, so the free cumulants of a sum are the sums of the free cumulants.

What would settle it

Compute the mixed free cumulants of an explicit pair of even-order tensor families that satisfy the paper's freeness definition; any nonzero mixed cumulant would invalidate the additivity $\kappa_n(\mu\oplus_p\nu)=\kappa_n(\mu)+\kappa_n(\nu)$ on which the convolution is built.

Watch

Extended reading notes

Core claim

The paper's main discovery is that the operation $\oplus_p$ is well defined on compactly supported measures that admit freely independent tensor-distribution lifts: for even $p$, the free cumulants of the sum add, $\kappa_n(\mu\oplus_p\nu)=\kappa_n(\mu)+\kappa_n(\nu)$ for $n\ge 1$, and the $R$-transform of the convolution is the sum of the $R$-transforms. The construction is carried by the analytic moment-cumulant formula $M_T(z)=C_T(z M_T(z)^{p/2})$ together with the vanishing of mixed free cumulants for even tensor families. The paper then computes the basic examples: the high-order semicircular law satisfies $\mu_p\oplus_p\mu_p = \mu_p^{(\sqrt{2})}$, a $\sqrt{2}$-dilated semicircular, and the high-order free Poisson laws satisfy $\nu_{p,t}\oplus_p\nu_{p,t'}=\nu_{p,t+t'}$. It also proves that a Wigner tensor converges to $\mu_p$, a Wishart-type tensor converges to $\nu_{p,t}$, and a free central limit theorem holds in this setting.

Load-bearing premise

The construction of $\oplus_p$ rests on an earlier theorem, not proved in this paper, that for even-order tensor families freeness is exactly the vanishing of all mixed free cumulants; if that theorem failed for the families being summed, the additivity that defines the convolution would collapse.

Editorial extensions

If this is right

  • The high-order semicircular law is stable under $\oplus_p$ up to dilation: $\mu_p\oplus_p\mu_p = \mu_p^{(\sqrt{2})}$.
  • Free Poisson parameters add: $\nu_{p,t}\oplus_p\nu_{p,t'}=\nu_{p,t+t'}$, and as $t\to\infty$ the reshaped free Poisson law tends to the high-order semicircular law.
  • For compactly supported measures, the $R$-transform of a $\oplus_p$-convolution is the sum of the $R$-transforms, giving a direct computational tool.
  • Wigner and Wishart tensors converge in probability to the high-order semicircular and free Poisson laws, and a free central limit theorem holds for even $p$.
  • A $Q$-transform, $Q_\mu(z)=(C_\mu(z)^{p/2}-1)/z$, is introduced as a tensorial substitute for the usual transform, with a subordination theory left for later work.

Reading between the lines

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

  • If the analytic machinery can be extended to odd $p$ by the multiplicity-corrected cumulants the paper sketches, the restriction to even $p$ in the convolution section could be lifted, giving a uniform $\oplus_p$ for all tensor orders.
  • The additive $R$-transform suggests that the standard free-probability toolbox, such as subordination functions and free multiplicative convolution, could be developed for tensor models and applied to spectral statistics of spiked or noisy tensors.
  • A direct numerical simulation of small even-order Wishart tensors (for instance $p=4$ with moderate $N$) should reproduce the Fuss-Narayana moments; such a simulation would test the convergence theorems in a regime the paper does not explore.
  • The $Q$-transform, which inverts the subordination function $G(z)=z^{p/2-1}g(z)^{p/2}$, may be the right object for studying sums of non-compactly supported tensor-valued random variables, a case the paper leaves open.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

5 major / 5 minor

Summary. The paper develops a tensorial analogue of free additive convolution in the framework of "tensorial freeness" introduced in the author's earlier work with Bordenave. It defines moments and free cumulants of a tensor as sums over connected p-regular trace maps, proves the analytic moment-cumulant formula M_T(z) = C_T(z M_T(z)^{p/2}) (Theorem 1), and introduces higher-order semicircular and free Poisson laws with moments given by Fuss-Catalan and Fuss-Narayana numbers. It also proves convergence of Wigner and Wishart tensors to these laws, states a free central limit theorem, and defines a tensorial free convolution \oplus_p, an R-transform, and a Q-transform, with additivity of the R-transform and examples for semicircular and free Poisson laws.

Significance. If the main results hold, the paper gives a substantial new extension of free convolution to the tensorial setting and connects several known higher-order laws. The combinatorial proof of Theorem 1 via Lemma 2 is a genuine strength: the moment-cumulant relation is derived from the poset rather than assumed, and the convergence results for Wigner and Wishart tensors are substantive. The identification of the semicircular and free Poisson laws with Fuss-Catalan and Fuss-Narayana objects is also useful. However, the construction of \oplus_p rests on a freeness criterion imported from [12] and on the exponential-boundedness argument in Proposition 5, which contains a false bound; the variance estimate in Proposition 4 is only sketched. These gaps concern the central claims and require repair.

major comments (5)
  1. [§4.1, Proposition 5] The proof of (ii)⇒(i) asserts |m_n(a)| ≤ F_p(n) M^n ≤ (2pM)^n. The second inequality is false for p≥3: the Fuss-Catalan numbers satisfy F_p(n) ∼ sqrt((p−1)/(2πp)) (p^p/(p−1)^{p−1})^n n^{-3/2}, and p^p/(p−1)^{p−1} > 2p for every p≥3 (e.g., for p=3 the base is 27/4 = 6.75 > 6). Since Proposition 6 relies on this proposition to conclude that the moments of a+b are exponentially bounded, the existence of the measure μ⊕_pν is not established by the argument as written. The result can likely be repaired by replacing 2p with the correct growth constant, but the paper must supply a correct bound.
  2. [§4.1, Proposition 5] The proof of (i)⇒(ii) is also incomplete: it bounds |κ_n(a)| by a sum involving the same unknown quantities |κ_{b_j}(a)| and then writes "≤ M^n" without an explicit induction hypothesis, and the final constant (4p)^n is not derived. This direction is needed in Proposition 6 to know that the free cumulants of μ and ν are exponentially bounded before adding them, so it must be made rigorous.
  3. [§3.3.2, Proposition 4] The variance estimate is only sketched and contains a non-obvious, unproved assertion: when G1 and G2 share a coincident edge and the graph of noncoincident edges has no cycle, the number of noncoincident vertices is claimed to be at most np via a parity/coincidence argument that is not spelled out. Since Var[m_n(W_N)] = O(N^{-2}) is the step that upgrades moment convergence to convergence in probability in Theorem 5, this gap is load-bearing; the proof should either be completed or replaced by a detailed counting argument.
  4. [§4.1, Proposition 6 and Definition 5] The convolution is defined only for pairs (μ,ν) for which there exist distributions a,b with μ_a=μ, μ_b=ν and a,b free, and Remark 4 concedes that not every distribution corresponds to a real tensor distribution. The paper does not show that all compactly supported measures, or even all measures arising from tensor distributions, admit such a free pair. Consequently the statements in the abstract and in Corollary 3 that the convolution and R-transform are defined for arbitrary compactly supported measures go beyond what is proved. Please either restrict the claims to the constructed class or prove a realization theorem.
  5. [§2.4, §4 opening, and Theorem 6] The additivity of free cumulants for sums and the characterization of freeness by vanishing mixed cumulants for even families are imported from [12] without stating the exact theorem and its hypotheses. Because the entire convolution construction in Section 4 and the free CLT in Theorem 6 depend on this criterion, the paper should state the precise result from [12] (or prove it in an appendix) so the reader can verify that the families of distributions used here satisfy it.
minor comments (5)
  1. [Throughout] There are several typographical errors: the heading "Propostion 3" in §3.3.1 should be "Proposition 3"; the abstract has "f reeness"; Remark 6 has "notataions" and "concernd"; §2.3 has "We remind thet Bn".
  2. [§3.3.1, proof of Proposition 3] In the final display, the summation index runs from b=0 to n−1, while F^b_{p/2}(n) is defined for 1≤b≤n and the moment formula in Definition 4 sums b=1,...,n; this appears to be a shift typo, and the limit should be ∑_{b=1}^n F^b_{p/2}(n) t_N^b + O(N^{-1}).
  3. [§4.3, Proposition 7] In item (2), "µ({a,b}/2" should read "µ({a,b})/2". In item (3), the support is denoted [−M,M], but the text says "for some R>0"; the notation should be made consistent.
  4. [§2.5, proof of Theorem 1] In the last display, the notation "M_T(s)^{sp/2}" should be "M_T(z)^{sp/2}"; the argument of the moment generating function is z, not the summation index s.
  5. [§4.1, after Proposition 6] The text says that "δ0 has all zero moments and free cumulants for n≥1", but then the distribution t.1_p is described with cumulants concentrated on bouquet maps and its measure Δ_t has cumulant t at n=1; the relation between δ0 and Δ_0 should be stated explicitly to avoid confusion.

Circularity Check

2 steps flagged · score 4.0 of 10

The convolution section inherits its additivity from the authors' previous work [12], and the R-transform/examples restate that additivity; the moment-cumulant and convergence arguments have independent content.

  1. self citation load bearing [Section 2.4 and Section 4 opening / Proposition 6]
    "The crucial point is that we proved that two even families of tensors of possibly different orders are free if and only if the mixed cumulants vanish. ... we fix p ≥ 2 an even integer, as the following proofs rely on the vanishing of mixed cumulants."

    The additivity used to form ⊕_p is not derived in this paper. Proposition 6 says 'By free independence of a and b, we know that for all n ≥ 1, κn(a + b) = κn(a) + κn(b)', which is exactly the content of [12] by the same authors. Since this vanishing-of-mixed-cumulants criterion is the only mechanism that makes κ_n additive, the existence of ⊕_p and all subsequent convolution identities rest on a load-bearing self-citation. If the criterion from [12] were unavailable or restricted differently, the construction in Section 4 would collapse; no independent proof, external check, or machine verification is supplied in the present text.

  2. self definitional [Section 4.1, Proposition 6 / Definition 5; Corollary 3; Lemmas 8-9]
    "By free independence of a and b, we know that for all n ≥ 1, κn(a + b) = κn(a) + κn(b). Hence, thanks to Proposition 5, we have that the moments of a + b are exponentially bounded. This gives the result. ... The probability measure µ ⊕p ν is called the tensorial free convolution of µ and ν."

    Definition 5 defines ⊕_p through the distribution of a+b for free representatives, so the additivity of free cumulants is built into the operation. Corollary 3's R-transform additivity is then the same relation rewritten with R=(C-1)/z, and Lemmas 8 and 9 merely insert the cumulant sequences (1,0,0,...) and (t,t,...) into that defining relation and solve. These results therefore reduce, by construction, to the cumulant additivity that defines the convolution; they check consistency rather than provide independent confirmation. The existence step retains independent combinatorial content, so the circularity is partial.

full rationale

Theorem 1 is proved from the non-crossing poset and Möbius inversion, so the analytic moment-cumulant relation has independent grounding. The Wishart convergence (Proposition 3) is proved by a detailed ∆-graph counting argument, and the Wigner convergence and free CLT are supported by the convergence lemmas and estimates; these are not circular. The circularity is confined to Section 4: the definition of ⊕_p relies on the even-family 'vanishing mixed cumulants' criterion imported from [12] (same authors), and the R-transform additivity plus the semicircular/Poisson convolution examples are immediate algebraic rewritings of that imported additivity. A separate correctness issue, not a circularity, is that Proposition 5's proof asserts F_p(n) ≤ (2p)^n, which is false for p ≥ 3; the existence argument would need a larger base or a different bound. Because the core moment-cumulant theorem and convergence results are independent of these definitional steps, the paper has substantial non-circular content; the convolution part is only as solid as the self-cited criterion, giving score 4.

Assumptions & free parameters 0 free parameters · 3 assumptions · 2 invented entities

The central claim rests on the author's prior freeness theory (vanishing mixed cumulants, asymptotic freeness), on Gurau's measure-existence theorem for real symmetric tensors, and on the known compact support of free Bessel laws. No parameters are fitted to data; the scaling factors in the Wigner and Wishart definitions are part of the model and are chosen to make limits nontrivial, rather than adjusted to match a target.

assumptions (3)
  • domain assumption The non-crossing poset on p-regular maps yields a Moebius-invertible moment-cumulant relation, and for even families freeness is equivalent to vanishing of mixed free cumulants.
    Invoked in Section 4 and in the proof of Theorem 6 as the basis for kappa_n(a+b) = kappa_n(a) + kappa_n(b); proved in the author's prior paper [12], not reproved here.
  • domain assumption For every real symmetric tensor T, there exists a probability measure mu_T on R with moments m_n(T).
    Used in Section 1.1 and Theorems 4 and 5 to speak of the measure of a Wigner or Wishart tensor; the paper explicitly notes that a direct proof is still missing and cites Gurau [21].
  • domain assumption The free Bessel laws pi_{p/2,t} exist, are compactly supported, and have Fuss-Narayana moments.
    Used in Definition 4 to assert that the high-order free Poisson law is compactly supported; cited from Banica et al. [8].
invented entities (2)
  • Tensorial free additive convolution oplus_p independent evidence
    purpose: Defines the law of the sum of two freely independent tensor distributions (or of measures admitting such distributions).
    Reduces to Voiculescu free convolution at p=2, has neutral element delta_0, is commutative and associative, and reproduces the semicircular and free Poisson additivity examples, so it is anchored to existing theory.
  • Tensorial R-transform and Q-transform independent evidence
    purpose: Power series encoding free cumulants, additive under oplus_p; Q is related to the formal inverse K(z) = C(z)^(p/2)/z.
    The R-transform reproduces the classical R-transform at p=2, and the Q-transform is derived from the proved relation K(G(z)) = z rather than being postulated in isolation.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Tensorial free convolution, semicircular, free Poisson and R-transform in high order." pith.science (2026). https://pith.science/paper/LO6H3QJV

@misc{pith2026241202572,
  author       = {Pith},
  title        = {Pith review of: Tensorial free convolution, semicircular, free Poisson and R-transform in high order},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LO6H3QJV}},
  note         = {Machine review of arXiv:2412.02572}
}
abstract

This work builds on our previous developments regarding a notion of freeness for tensors. We aim to establish a tensorial free convolution for compactly supported measures. First, we define higher-order analogues of the semicircular (or Wigner) law and the free Poisson (or Marcenko-Pastur) law, giving their moments and free cumulants. We prove the convergence of a Wishart-type tensor to the free Poisson law and recall the convergence of a Wigner tensor to the semicircular law. We also present a free Central Limit Theorem in this context. Next, we introduce a tensorial free convolution, define the $R$-transform, and provide the first examples of free convolution of measures.

Figures

Figures reproduced from arXiv: 2412.02572 by the authors.

Figure 1
Figure 1. Some tensor maps. α = (σ(1), σ(2))· · · (σ(p − 1), σ(p)). The canonical one is b id p . It is a particular case of multicycle map, that is n ≥ 1 vertices p/2 edges between the i-th and the i + 1-th ones for all 1 ≤ i ≤ n (n + 1 = 1). For a given choice of inputs and outputs edges, there are [(p/2)!]n such maps. For p odd, an odd multicycle is a map with 2n vertices and (p + 1)/2 edges between the 2i − 1-th and the 2… view at source ↗
Figure 2
Figure 2. Poset Pπ We remind thet Bn is the set of connected rooted p-regular combinatorial maps in M0 with n (unlabeled) vertices. Definition 1 (Moments of a tensor). For T a tensor of order p, we call mn(T) := X b∈Bn b(T) the n-th moment of T. By convention we set m0(T) = 1. Finally, we define the generating function of these moments, for z ∈ C, MT (z) := X n≥0 mn(T)z n , as a formal power series in C[[z]]. 2.4 Free cumulan… view at source ↗
Figure 3
Figure 3. T and T σ . If p1, p2 are two integers, we denote Pp1,p2 the set of application π : [[p1 + p2]] 7→ {1, 2}×[[p1 + p2]] such that |π −1 1 (1)| = p1, |π −1 1 (2)| = p1 and π2 is injective. We write π −1 (1) the ordered p1- tuple of elements in π −1 1 (1) ordered by their image by π, that is π −1 (1) = (x1, . . . , xp1 ) such that 13 [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: A ∆-graph (p = 4, n = 3, b = 2, r + 1 = 5). Two graphs are said isomorphic if there are the same up to a permutation on (1, . . . , k) and a permutation on (1, . . . , N). For each isomorphism class, there is only one canonical graph satisfying i1 = j1 = 1, iu+1 ≤ max{…
Figure 5
Figure 5. Figure 5: A ∆1(n, b)-graph. combinatorics. Lemma 3. For given n, b, r, the number of graphs in the isomorphism class of G ∈ ∆(n, b, r) is k(k − 1). . .(k − b + 1)N(N − 1). . .(N − r)c G p = k bN r+1c G p (1 + O(N −1 ). where c G p is a constant depending only on the shape of G, …

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

36 extracted references · 35 canonical work pages

  1. [12]

    Bonnin and C

    R. Bonnin and C. Bordenave. Freeness for tensors, 2024

    work page 2024

  2. [1]

    G. W. Anderson, A. Guionnet, and O. Zeitouni. An introduction to random matrices , volume 118 of Cambridge Studies in Advanced Mathematics . Cambridge University Press, Cambridge, 2010

    work page 2010

  3. [2]

    G. B. Arous, S. Mei, A. Montanari, and M. Nica. The landsca pe of the spiked tensor model, 2018

    work page 2018

  4. [3]

    B. Au, G. C´ ebron, A. Dahlqvist, F. Gabriel, and C. Male. F reeness over the diagonal for large random matrices. Annals of Probability , 49:157–179, 01 2021

    work page 2021

  5. [4]

    R. C. Avohou, J. B. Geloun, and R. Toriumi. Counting u(n)⊗r ⊗ o(n)⊗q invariants and tensor model observables, 2024

    work page 2024

  6. [5]

    Bai and J

    Z. Bai and J. W. Silverstein. Spectral Analysis of Large Dimensional Random Matrices . 2010

    work page 2010

  7. [6]

    A. S. Bandeira, M. T. Boedihardjo, and R. van Handel. Matr ix concentration inequalities and free probability. Inventiones mathematicae, 234(1):419–487, June 2023

    work page 2023

  8. [7]

    A. S. Bandeira, S. Gopi, H. Jiang, K. Lucca, and T. Rothvos s. A geometric perspective on the injective norm of sums of random tensors, 2024

    work page 2024

Show all 36 references
  1. [8]

    Banica, S

    T. Banica, S. T. Belinschi, M. Capitaine, and B. Collins. Free bessel laws. Canadian Journal of Mathematics , 63(1):3–37, Feb. 2011

    2011

  2. [9]

    S. T. Belinschi, T. Mai, and R. Speicher. Analytic subord ination theory of operator-valued free additive convolution and the solution of a general rand om matrix problem. Journal f¨ ur die reine und angewandte Mathematik (Crelles Journal) , 2017(732):21–53, 2017

    2017

  3. [10]

    Benedetti and N

    D. Benedetti and N. Delporte. Remarks on a melonic field t heory with cubic interaction. Journal of High Energy Physics , 2021(4), Apr. 2021. 29

    2021

  4. [11]

    R. Bonnin. Universality of the wigner-gurau limit for r andom tensors, 2024

    2024

  5. [13]

    Bordenave and D

    C. Bordenave and D. Chafai. Around the circular law. Probability Surveys, 9, 09 2011

    2011

  6. [14]

    Capitaine and C

    M. Capitaine and C. Donati-Martin. Spectrum of deforme d random matrices and free proba- bility, 2016

    2016

  7. [15]

    J. Cima, A. Matheson, and W. Ross. The Cauchy Transform. American Mathematical Society, 10 2006

    2006

  8. [16]

    Collins, R

    B. Collins, R. Gurau, and L. Lionni. Free cumulants and f reeness for unitarily invariant random tensors, 2024

    2024

  9. [17]

    Collins, J

    B. Collins, J. A. Mingo, P. Sniady, and R. Speicher. Seco nd order freeness and fluctuations of random matrices. iii: Higher order freeness and free cumu lants. Documenta Mathematica, 12:1–70, 2007

    2007

  10. [18]

    Collins and I

    B. Collins and I. Nechita. Random matrix techniques in q uantum information theory. Journal of Mathematical Physics , 57, 09 2015

    2015

  11. [19]

    P. H. Edelman. Chain enumeration and non-crossing part itions. Discrete Mathematics , 31(2):171–180, 1980

    1980

  12. [20]

    R. Gurau. Random tensors. Oxford: Oxford University Press, 2017

    2017

  13. [21]

    R. Gurau. On the generalization of the wigner semicircl e law to real symmetric tensors, 2020

    2020

  14. [22]

    Gurau and V

    R. Gurau and V. Rivasseau. Quantum gravity and random te nsors. arXiv:2401.13510, 2024

    2024 arXiv

  15. [23]

    Jagannath, P

    A. Jagannath, P. Lopatto, and L. Miolane. Statistical t hresholds for tensor pca. The Annals of Applied Probability , 30(4), Aug. 2020

    2020

  16. [24]

    Kunisky, C

    D. Kunisky, C. Moore, and A. S. Wein. Tensor cumulants fo r statistical inference on invariant distributions, 2024

    2024

  17. [25]

    Marˇ cenko and L

    V. Marˇ cenko and L. Pastur. Distribution of eigenvalue s for some sets of random matrices. Math USSR Sb , 1:457–483, 01 1967

    1967

  18. [26]

    J. A. Mingo and A. Nica. Annular noncrossing permutatio ns and partitions, and second-order asymptotics for random matrices. International Mathematics Research Notices, 2004(28):1413– 1460, 01 2004. 30

    2004

  19. [27]

    J. A. Mingo and R. Speicher. Free probability and random matrices , volume 35 of Fields Inst. Monogr. Toronto: The Fields Institute for Research in the Mathemati cal Sciences; New York, NY: Springer, 2017

    2017

  20. [28]

    Nica and R

    A. Nica and R. Speicher. Lectures on the Combinatorics of Free Probability Theory. Cambridge: Cambridge University Press, 2006

    2006

  21. [29]

    Pastur and M

    L. Pastur and M. Shcherbina. Eigenvalue Distribution of Large Random Matrices . Mathemat- ical Surveys and Monographs, 2011

    2011

  22. [30]

    Speicher

    R. Speicher. Multiplicative functions on the lattice o f non-crossing partitions and free convo- lution. Mathematische Annalen , 298(4):611–628, 1994

    1994

  23. [31]

    Speicher

    R. Speicher. Lecture notes on ”free probability theory ”, 2019

    2019

  24. [32]

    Voiculescu

    D. Voiculescu. Addition of certain non-commuting rand om variables. Journal of Functional Analysis, 66(3):323–346, 1986

    1986

  25. [33]

    Voiculescu

    D. Voiculescu. Limit laws for random matrices and free p roducts. Invent. Math. , 104(1):201– 220, 1991

    1991

  26. [34]

    Voiculescu

    D. Voiculescu. A hydrodynamic exercise in free probabi lity: Setting up free euler equations, 2019

    2019

  27. [35]

    Voiculescu, K

    D. Voiculescu, K. J. Dykema, and A. Nica. Free random variables , volume 1 of CRM Mono- graph Series . American Mathematical Society, Providence, RI, 1992. A no ncommutative probability approach to free products with applications to random matrices, operator algebras and harmo...

    1992

  28. [36]

    E. P. Wigner. On the distribution of the roots of certain symmetric matrices. Annals of Mathematics, 67(2):325–327, 1958. 31

    1958

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.