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The large $N$ factorization does not hold for arbitrary multi-trace observables in random tensors

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Pith's one-line read Gaussian random tensors of order three or higher fail to factorize multi-trace expectations at large N, and the failure is typical among large observables.

desk verdict Solid, important counterexample: large-N factorization fails for Gaussian random tensors in D>=3; the proof is sound despite a Stirling typo and a compressed lemma. read the letter →

arxiv 2506.15362 v1 pith:U2F6U3A2 submitted 2025-06-18 math-ph hep-thmath.COmath.MPmath.PR

Razvan Gurau , Felix Joos , Benjamin Sudakov This is my paper · ORCID
classification math-phhep-thmath.COmath.MPmath.PR MSC 60B2005C8005C70
keywords randomtensorslargeNfactorizationtraceinvariantsGaussianmelonicgraphsstrictsubadditivityclassicalcumulantsedgeD-colored
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 proves that for Gaussian random tensors of order $D \ge 3$, expectations of products of trace invariants do not in general factorize at large $N$ into products of single-trace expectations, contrary to a conjecture in the tensor free probability literature. The root cause is that the Gaussian scaling of joint cumulants is not strictly subadditive: there exist observables for which the connected two-point expectation scales like $N^{Dn}$, strictly larger than the square of the one-point expectation. This marks a sharp contrast with random matrices ($D=2$), where factorization always holds. The paper further shows that factorization does survive for the melonic observables, the dominant family in the large-$N$ limit.

What carries the argument

The central objects are edge $D$-colored graphs: graphs on $2n$ vertices whose edges are partitioned into $D$ perfect matchings, one per tensor-index color. Each such graph $M$ carries a trace invariant $\mathrm{Tr}_M(T)$, and the Gaussian expectation of a product of invariants becomes a sum over pairings $M_0$ of tensor entries, with each term contributing $N^{F_n(M_0,M)}$, where $F_n(M_0,M)$ counts the cycles that alternate between the pairing $M_0$ and the colored matchings. The Gaussian scaling of an observable is $\max_{M_0} F_n(M_0,M)$. Lemma 1 reduces strict subadditivity of this scaling (equivalently, large-$N$ factorization) to the component-wise bound $\max_{M_0} F_n(M_0,M) > Dn/2$ for every connected graph; its converse uses boundary graphs, which compress the information contained in a partial pairing. The proof then runs a probabilistic argument: a tail bound for the number of alternating cycles of two random matchings (Proposition 2) shows that for a fixed pairing $M_0$, the $D$ random matchings have total $F_n(M_0,M) < (1+o(1))n$ with probability close to 1, and a union bound over all $M_0$ yields a graph violating the bound for every $M_0$.

What would settle it

For the smallest $D=3$ graphs produced by the probabilistic construction, compute the exact leading power of $N$ in the connected expectation $\langle \mathrm{Tr}_M(T)\mathrm{Tr}_M(T)\rangle_{\mathrm{con}}$ and compare it with $2 \max_{M_0} F_n(M_0,M)$; if it fails to exceed that product, the claimed non-factorization mechanism would be wrong. More sharply, any single connected graph violating $\max_{M_0} F_n(M_0,M) > Dn/2$ while still obeying strict subadditivity would disprove the equivalence in Lemma 1 on which the theorem depends.

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Extended reading notes

Core claim

For every $D \ge 3$ and every $\epsilon > 0$, for $n$ large enough, a fraction at least $1-\epsilon$ of all edge $D$-colored graphs $M$ on $2n$ vertices have a connected component $M_\rho$ with $2n_\rho$ vertices such that $\max_{M_0 \in \mathcal{M}_{2n_\rho}} F_{2n_\rho}(M_0, M_\rho \sqcup M_\rho) \ge D n_\rho$, while $2 \max_{M_0 \in \mathcal{M}_{n_\rho}} F_{n_\rho}(M_0, M_\rho) < D n_\rho$. Consequently the connected expectation $\langle \mathrm{Tr}_{M_\rho}(T)\,\mathrm{Tr}_{M_\rho}(T)\rangle_{\mathrm{con}}$ scales at least as $N^{D n_\rho}$, strictly larger than the product $\langle \mathrm{Tr}_{M_\rho}(T)\rangle\,\langle \mathrm{Tr}_{M_\rho}(T)\rangle$. In other words, the Gaussian scaling of cumulants is not strictly subadditive for $D \ge 3$, so large-$N$ factorization of arbitrary multi-trace expectations fails for tensors, while it holds for the melonic family and for $D=2$.

Load-bearing premise

The proof of the main theorem rests on the converse direction of Lemma 1: strict subadditivity of the Gaussian scaling is equivalent to the bound $\max_{M_0} F_n(M_0,M) > Dn/2$ holding for every connected graph $M$; that direction relies on the claim that a partial pairing of boundary graphs can always be re-paired component-wise to produce more than $Ds/2$ alternating cycles, and if this re-pairing claim fails, the violation of the bound would not imply non-factorization.

Editorial extensions

If this is right

  • The conjecture that Gaussian random tensors satisfy large-$N$ factorization, proposed in the tensor free probability literature, is false for $D \ge 3$.
  • Large-$N$ factorization survives for the melonic family of observables, so the leading large-$N$ sector of tensor models still admits a factorization description.
  • For $D \ge 3$, generic multi-trace observables can have connected expectations that dominate the product of single-trace expectations, making the large-$N$ structure of random tensor models richer than that of random matrices.
  • The result leaves untouched the known factorization of matrix moments for $D=2$.
  • The proof identifies a sharp threshold: non-factorization appears once the Gaussian scaling bound for a connected graph with $2n$ vertices falls at or below $Dn/2$.

Reading between the lines

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

  • The proof suggests that non-factorizing observables are generic rather than exceptional: for large $n$, almost every graph violates the subadditivity bound, so the failure should persist for any Gaussian tensor ensemble with the same Wick structure.
  • The same combinatorial mechanism should carry over to complex Gaussian tensors and to tensors with $O(N)^D$ symmetry, since the argument only uses the pairing structure of the Gaussian covariance, not the realness of entries.
  • For proposed tensorial generalizations of free probability, the failure of factorization implies that asymptotic moment–free cumulant relations derived under a factorization assumption would receive corrections at subleading orders; the counterexample family gives a concrete place to test such corrections.
  • A testable extension: the typical value of $\max_{M_0} F_n(M_0,M)$ for random $D$-colored graphs may be close to $(1+o(1))n$, and refining the tail bound could give the exact distribution of non-factorization exponents.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 6 minor

Summary. The paper studies Gaussian random tensors of order D >= 3 and the large-N scaling of connected expectations of multi-trace invariants. It proves that the Gaussian scaling is not strictly subadditive: for n large, almost every edge D-colored graph M on 2n vertices contains a connected component M_rho with 2n_rho vertices such that the connected expectation <Tr_{M_rho}(T) Tr_{M_rho}(T)>_con has scaling exponent at least D n_rho, while the product <Tr_{M_rho}(T)>^2 has scaling exponent strictly smaller. The proof uses the probabilistic method: Proposition 2 bounds the number of alternating cycles between a fixed and a random perfect matching, and Proposition 3 shows that for almost every D-tuple of random perfect matchings the maximal cycle count is (1+o(1))n, hence below Dn/2 for D >= 3. The paper also verifies that the melonic family does satisfy large-N factorization, contrasting with the general failure.

Significance. If the result stands, it refutes a conjecture in the random-tensor/free-probability literature and establishes a sharp qualitative separation from random matrices, where multi-trace expectations always factor at large N. The argument is self-contained, uses no fitted parameters, and gives explicit quantitative bounds on the fraction of counterexamples; Propositions 2 and 3 are proven in detail and the main theorem follows from them by a direct copy-pairing argument that bypasses the more speculative converse of Lemma 1. This is a compact but substantial negative result that will be of interest to the random tensor and physics communities.

minor comments (6)
  1. [Section 3, displayed bound before Proposition 3] The displayed upper bound on the number of perfect matchings is numerically false as written: it states |M_n| < sqrt(2e) 2^n e^n / n^n, but for n=4 this gives about 7.95 while the true value is |M_4|=105. The intended bound is |M_n| < sqrt(2e) 2^n n^n e^{-n}, and the subsequent union bound in Proposition 3 uses this corrected form. Please fix the display and the surrounding Stirling estimate.
  2. [Section 3, Lemma 1 proof] The sufficiency direction of Lemma 1 is only sketched. The key step asserts that re-pairing each boundary graph produces more than Ds/2 alternating cycles and that this implies F_n(M0,M) < sum_rho F_{n_rho}(M0_rho union M0'_rho, M_rho), but the counting argument behind this claim is not given. Since Theorem 1 is subsequently proved directly in the last subsection via the copy-pairing of a component from Proposition 3, this gap does not affect the main theorem, but the lemma should either be proved in full or stated with only the necessary direction and a remark that the direct argument is used.
  3. [Section 3, Proposition 3 proof] The line beginning with 'n^D prod_{t1+t2+...+tD=A}' is not well-formed; after choosing the maximizing tuple and using independence, the bound should be written as n^D (7^n)^D (2n)^{-A}. Please correct this display.
  4. [Section 2, Eq. (2.5)] In the right-hand side of (2.5), the index of F is written as F_{n1}(M0_rho, M_rho); it should be F_{n_rho}(M0_rho, M_rho).
  5. [Section 3, last paragraph] The sentence 'for D=3 for example, starting at an n of order ~7 6' is unclear; the exponent appears to be missing and the exact threshold for the inequality (D-2)n/2 > (D ln 7 - ln 2)n/ln n + D should be stated explicitly.
  6. [Throughout] There are several small typographical slips, such as 'connect M' instead of 'connected M' shortly after Eq. (2.4), and the repeated use of n1 instead of n_rho in the proof of Lemma 1. These should be cleaned up in revision.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the counterexample is derived from a self-contained probabilistic construction and a direct scaling comparison, with no fitted parameter or conclusion assumed.

full rationale

Theorem 1 is not circular. The Gaussian expectation formula (2.2) and the cumulant expansion (2.4) follow directly from Wick's theorem and Möbius inversion; Proposition 2 is proved by an explicit induction on random matchings; Proposition 3 is a union-bound argument over the finitely many matchings M0. The final step is a direct double-trace comparison: Proposition 3 gives a graph M with F_n(M0,M) < Dn/2 for every M0. If M is disconnected, a union-of-optimal-matchings argument forces some component Mρ to have max F_{nρ} < Dnρ/2. Pairing the two copies of Mρ by the copy-wise matching yields a connected G(M0, Mρ⊔Mρ) with F = Dnρ alternating cycles, so the connected expectation scales at least as N^{Dnρ}, strictly larger than the product scaling 2 max F_{nρ} < Dnρ. This argument bypasses Lemma 1, so any concern about the compressed converse direction of Lemma 1 is a correctness-risk issue, not circularity. The only self-references are contextual: the conjecture from [41] is the statement being refuted, and Proposition 1 from earlier work is used only for the auxiliary melonic-family verification, not for the negative result.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters: the Gaussian variance scale nu is conventional and the results are independent of it; the proof introduces only mathematical constructions (boundary graphs) as proof tools, not new physical entities. All inputs are standard combinatorial and probabilistic facts.

assumptions (6)
  • standard math Wick's theorem for Gaussian integration, giving the sum over perfect matchings for expectations of products of tensor entries.
    Used in Section 2 to write the Gaussian expectation of trace invariants as a sum over pairings M0.
  • standard math Stirling's approximation for factorials and the resulting bound on the number of perfect matchings on 2n elements.
    Used in Section 2 and in Proposition 3 to count matchings and perform union bounds.
  • standard math Markov's inequality for nonnegative random variables.
    Used in the proof of Proposition 2 to convert the moment bound into a tail bound.
  • standard math Mobius inversion over the lattice of set partitions, defining classical cumulants from expectations.
    Used in Section 2 to introduce connected expectations and the factorization criterion.
  • domain assumption Domain assumption: tensor entries are independent identically distributed centered Gaussians with variance N^{-nu}.
    This is the model under study (Section 2, Gaussian random tensors); the result is stated for this measure.
  • standard math Euler relation for ribbon graphs (only in the planar side remark).
    Used in Section 3 to derive a bound for planar observables; not needed for the main theorem.

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Pith. "Pith review of The large $N$ factorization does not hold for arbitrary multi-trace observables in random tensors." pith.science (2026). https://pith.science/paper/U2F6U3A2

@misc{pith2026250615362,
  author       = {Pith},
  title        = {Pith review of: The large $N$ factorization does not hold for arbitrary multi-trace observables in random tensors},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/U2F6U3A2}},
  note         = {Machine review of arXiv:2506.15362}
}
abstract

We consider real tensors of order $D$, that is $D$-dimensional arrays of real numbers $T_{a^1a^2 \dots a^D}$, where each index $a^c$ can take $N$ values. The tensor entries $T_{a^1a^2 \dots a^D}$ have no symmetry properties under permutations of the indices. The invariant polynomials built out of the tensor entries are called trace invariants. We prove that for a Gaussian random tensor with $D\ge 3$ indices (that is such that the entries $T_{a^1a^2 \dots a^D}$ are independent identically distributed Gaussian random variables) the cumulant, or connected expectation, of a product of trace invariants is not always suppressed in scaling in $N$ with respect to the product of the expectations of the individual invariants. Said otherwise, not all the multi-trace expectations factor at large $N$ in terms of the single-trace ones and the Gaussian scaling is not subadditive on the connected components. This is in stark contrast to the $D=2$ case of random matrices in which the multi-trace expectations always factor at large $N$. The best one can do for $D\ge 3$ is to identify restricted families of invariants for which the large $N$ factorization holds and we check that this indeed happens when restricting to the family of melonic observables, the dominant family in the large $N$ limit.

Figures

Figures reproduced from arXiv: 2506.15362 by the authors.

Figure 1
Figure 1. Some melonic graphs with three colors. Left: the unique graph with two vertices [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. The first and the third graph display the same edge 3-colored graph with two [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

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Reference graph

Works this paper leans on

53 extracted references · 50 canonical work pages · cited by 1 Pith paper

  1. [1]

    Three-dimensional simplicial quantum gravity and generalized matrix models,

    J. Ambjorn, B. Durhuus, and T. Jonsson, “Three-dimensional simplicial quantum gravity and generalized matrix models,”Mod. Phys. Lett., vol. A6, pp. 1133–1146, 1991. 12

    work page 1991

  2. [2]

    Tensor model for gravity and orientability of manifold,

    N. Sasakura, “Tensor model for gravity and orientability of manifold,”Mod. Phys. Lett., vol. A6, pp. 2613–2624, 1991

    work page 1991

  3. [3]

    Gurau,Random Tensors

    R. Gurau,Random Tensors. Oxford: Oxford University Press, 2016

    work page 2016

  4. [4]

    Critical behavior of colored tensor models in the largeNlimit,

    V. Bonzom, R. Gurau, A. Riello, and V. Rivasseau, “Critical behavior of colored tensor models in the largeNlimit,”Nucl. Phys., vol. B853, pp. 174–195, 2011

    work page 2011

  5. [5]

    The 1/Nexpansion of the symmetric traceless and the antisymmetric tensor models in rank three,

    D. Benedetti, S. Carrozza, R. Gurau, and M. Kolanowski, “The 1/Nexpansion of the symmetric traceless and the antisymmetric tensor models in rank three,”Commun. Math. Phys., vol. 371, no. 1, pp. 55–97, 2019

    work page 2019

  6. [6]

    Uncolored Random Tensors, Melon Diagrams, and the SYK Models,

    I. R. Klebanov and G. Tarnopolsky, “Uncolored Random Tensors, Melon Diagrams, and the SYK Models,”Phys. Rev., vol. D95, no. 4, p. 046004, 2017

    work page 2017

  7. [7]

    LargeNlimit of irreducible tensor models:O(N) rank-3 tensors with mixed permutation symmetry,

    S. Carrozza, “LargeNlimit of irreducible tensor models:O(N) rank-3 tensors with mixed permutation symmetry,”JHEP, vol. 06, p. 039, 2018

    work page 2018

  8. [8]

    Melonic LargeNLimit of 5-Index Irreducible Random Tensors,

    S. Carrozza and S. Harribey, “Melonic LargeNLimit of 5-Index Irreducible Random Tensors,” Commun. Math. Phys., vol. 390, no. 3, pp. 1219–1270, 2022

    work page 2022

Show all 53 references
  1. [9]

    Melonic Dominance in Subchromatic Sextic Tensor Models,

    S. Prakash and R. Sinha, “Melonic Dominance in Subchromatic Sextic Tensor Models,”Phys. Rev. D, vol. 101, no. 12, p. 126001, 2020

    2020

  2. [10]

    Gapless spin fluid ground state in a random, quantum Heisenberg magnet,

    S. Sachdev and J. Ye, “Gapless spin fluid ground state in a random, quantum Heisenberg magnet,”Phys. Rev. Lett., vol. 70, p. 3339, 1993

    1993

  3. [11]

    A simple model of quantum holography,

    A. Kitaev, “A simple model of quantum holography,”KITP strings seminar and Entanglement 2015, Feb. 12, April 7, and May 27, 2015

    2015

  4. [12]

    Remarks on the Sachdev-Ye-Kitaev model,

    J. Maldacena and D. Stanford, “Remarks on the Sachdev-Ye-Kitaev model,”Phys. Rev., vol. D94, no. 10, p. 106002, 2016

    2016

  5. [13]

    An SYK-Like Model Without Disorder,

    E. Witten, “An SYK-Like Model Without Disorder,”J. Phys. A, vol. 52, no. 47, p. 474002, 2019

    2019

  6. [14]

    Phases Of Melonic Quantum Mechanics,

    F. Ferrari and F. I. Schaposnik Massolo, “Phases Of Melonic Quantum Mechanics,”Phys. Rev. D, vol. 100, no. 2, p. 026007, 2019

    2019

  7. [15]

    Spectrum of Majorana Quantum Mechanics withO(4) 3 Symmetry,

    K. Pakrouski, I. R. Klebanov, F. Popov, and G. Tarnopolsky, “Spectrum of Majorana Quantum Mechanics withO(4) 3 Symmetry,”Phys. Rev. Lett., vol. 122, no. 1, p. 011601, 2019

    2019

  8. [16]

    Spectra of eigenstates in fermionic tensor quantum mechanics,

    I. R. Klebanov, A. Milekhin, F. Popov, and G. Tarnopolsky, “Spectra of eigenstates in fermionic tensor quantum mechanics,”Phys. Rev., vol. D97, no. 10, p. 106023, 2018

    2018

  9. [17]

    Bosonic tensor models at largeNand small ϵ,

    S. Giombi, I. R. Klebanov, and G. Tarnopolsky, “Bosonic tensor models at largeNand small ϵ,”Phys. Rev., vol. D96, no. 10, p. 106014, 2017

    2017

  10. [18]

    Prismatic LargeN Models for Bosonic Tensors,

    S. Giombi, I. R. Klebanov, F. Popov, S. Prakash, and G. Tarnopolsky, “Prismatic LargeN Models for Bosonic Tensors,”Phys. Rev., vol. D98, no. 10, p. 105005, 2018

    2018

  11. [19]

    Symmetry Breaking in Coupled SYK or Tensor Models,

    J. Kim, I. R. Klebanov, G. Tarnopolsky, and W. Zhao, “Symmetry Breaking in Coupled SYK or Tensor Models,”Phys. Rev. X, vol. 9, no. 2, p. 021043, 2019

    2019

  12. [20]

    Notes on melonic O(N)q−1 tensor models,

    S. Choudhury, A. Dey, I. Halder, L. Janagal, S. Minwalla, and R. Poojary, “Notes on melonic O(N)q−1 tensor models,”JHEP, vol. 06, p. 094, 2018. 13

    2018

  13. [21]

    Phase diagram and fixed points of tensorial Gross-Neveu models in three dimensions,

    D. Benedetti and N. Delporte, “Phase diagram and fixed points of tensorial Gross-Neveu models in three dimensions,”JHEP, vol. 01, p. 218, 2019

    2019

  14. [22]

    Remarks on a melonic field theory with cubic interaction,

    D. Benedetti and N. Delporte, “Remarks on a melonic field theory with cubic interaction,” JHEP, vol. 04, p. 197, 2021

    2021

  15. [23]

    Line of fixed points in a bosonic tensor model,

    D. Benedetti, R. Gurau, and S. Harribey, “Line of fixed points in a bosonic tensor model,” JHEP, vol. 06, p. 053, 2019

    2019

  16. [24]

    Notes on tensor models and tensor field theories,

    R. G. Gurau, “Notes on tensor models and tensor field theories,”Ann. Inst. H. Poincare D Comb. Phys. Interact., vol. 9, no. 1, pp. 159–218, 2022

    2022

  17. [25]

    TASI Lectures on LargeNTensor Models,

    I. R. Klebanov, F. Popov, and G. Tarnopolsky, “TASI Lectures on LargeNTensor Models,” PoS, vol. TASI2017, p. 004, 2018

    2018

  18. [26]

    Melonic CFTs,

    D. Benedetti, “Melonic CFTs,”PoS, vol. CORFU2019, p. 168, 2020

    2020

  19. [27]

    On the generalization of the Wigner semicircle law to real symmetric tensors,

    R. Gurau, “On the generalization of the Wigner semicircle law to real symmetric tensors,” 4

  20. [28]

    Melonic dominance and the largest eigenvalue of a large random tensor,

    O. Evnin, “Melonic dominance and the largest eigenvalue of a large random tensor,”Lett. Math. Phys., vol. 111, no. 3, p. 66, 2021

    2021

  21. [29]

    Three Cases of Complex Eigenvalue/Vector Distributions of Symmetric Order-Three Random Tensors,

    S. Majumder and N. Sasakura, “Three Cases of Complex Eigenvalue/Vector Distributions of Symmetric Order-Three Random Tensors,”PTEP, vol. 2024, no. 9, p. 093A01, 2024

    2024

  22. [30]

    The edge of random tensor eigenvalues with deviation,

    N. Delporte and N. Sasakura, “The edge of random tensor eigenvalues with deviation,”JHEP, vol. 01, p. 071, 2025

    2025

  23. [31]

    Usefulness of signed eigenvalue/vector distributions of random tensors,

    M. R. Kloos and N. Sasakura, “Usefulness of signed eigenvalue/vector distributions of random tensors,”Lett. Math. Phys., vol. 114, no. 3, p. 80, 2024

    2024

  24. [32]

    Universality of the Wigner-Gurau limit for random tensors,

    R. Bonnin, “Universality of the Wigner-Gurau limit for random tensors,” 4 2024. arxiv: 2404.14144

    2024

  25. [33]

    The number of eigenvalues of a tensor,

    D. Cartwright and B. Sturmfels, “The number of eigenvalues of a tensor,”Linear algebra and its applications, vol. 438, no. 2, pp. 942–952, 2013

    2013

  26. [34]

    How many eigenvalues of a random symmetric tensor are real?,

    P. Breiding, “How many eigenvalues of a random symmetric tensor are real?,”Transactions of the American Mathematical Society, vol. 372, no. 11, pp. 7857–7887, 2019

    2019

  27. [35]

    The landscape of the spiked tensor model,

    G. B. Arous, S. Mei, A. Montanari, and M. Nica, “The landscape of the spiked tensor model,” Communications on Pure and Applied Mathematics, vol. 72, no. 11, pp. 2282–2330, 2019

    2019

  28. [36]

    Limit laws for random matrices and free products,

    D. Voiculescu, “Limit laws for random matrices and free products,”Invent. Math., vol. 104, p. 201, 1991

    1991

  29. [37]

    Voiculescu, K

    D. Voiculescu, K. Dykema, and A. Nica,Free random variables. AMS: CRM Monograph Series, 1992

    1992

  30. [38]

    The tensor Harish-Chandra–Itzykson–Zuber integral I: Weingarten calculus and a generalization of monotone Hurwitz numbers,

    B. Collins, R. Gurau, and L. Lionni, “The tensor Harish-Chandra–Itzykson–Zuber integral I: Weingarten calculus and a generalization of monotone Hurwitz numbers,”J. Eur. Math. Soc., vol. 26, no. 5, pp. 1851–1897, 2024

    2024

  31. [39]

    The Tensor Harish-Chandra–Itzykson–Zuber Integral II: Detecting Entanglement in Large Quantum Systems,

    B. Collins, R. Gurau, and L. Lionni, “The Tensor Harish-Chandra–Itzykson–Zuber Integral II: Detecting Entanglement in Large Quantum Systems,”Commun. Math. Phys., vol. 401, no. 1, pp. 669–716, 2023. 14

    2023

  32. [40]

    Freeness for tensors,

    R. Bonnin and C. Bordenave, “Freeness for tensors,” 7 2024. arXiv :2407.18881

    2024 arXiv

  33. [41]

    Free cumulants and freeness for unitarily invariant random tensors,

    B. Collins, R. Gurau, and L. Lionni, “Free cumulants and freeness for unitarily invariant random tensors,” 10 2024. arXiv: 2410.00908

    2024 arXiv

  34. [42]

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

    R. Bonnin, “Tensorial free convolution, semicircular, free Poisson and R-transform in high order,” 12 2024. arxiv: 2412.02572

    2024 arXiv

  35. [43]

    Spectral asymptotics for contracted tensor ensembles,

    B. Au and J. Garza-Vargas, “Spectral asymptotics for contracted tensor ensembles,”Electronic Journal of Probability, vol. 28, 01 2023

    2023

  36. [44]

    Nica and R

    A. Nica and R. Speicher,Lectures on the combinatorics of free probability (London Mathematical Society Lecture Note Series). Cambridge: Cambridge University Press, 2006

    2006

  37. [45]

    The LargeNlimit of superconformal field theories and supergravity,

    J. M. Maldacena, “The LargeNlimit of superconformal field theories and supergravity,”Adv. Theor. Math. Phys., vol. 2, pp. 231–252, 1998

    1998

  38. [46]

    Holography from Conformal Field Theory,

    I. Heemskerk, J. Penedones, J. Polchinski, and J. Sully, “Holography from Conformal Field Theory,”JHEP, vol. 10, p. 079, 2009

    2009

  39. [47]

    Tensor networks for complex quantum systems,

    R. Or´ us, “Tensor networks for complex quantum systems,”APS Physics, vol. 1, pp. 538–550, 2019

    2019

  40. [48]

    Counting tensor model observables and branched covers of the 2-sphere,

    J. Ben Geloun and S. Ramgoolam, “Counting tensor model observables and branched covers of the 2-sphere,”Ann. Inst. H. Poincare D Comb. Phys. Interact., vol. 1, no. 1, pp. 77–138, 2014

    2014

  41. [49]

    Tensor Models, Kronecker coefficients and Permutation Centralizer Algebras,

    J. Ben Geloun and S. Ramgoolam, “Tensor Models, Kronecker coefficients and Permutation Centralizer Algebras,”JHEP, vol. 11, p. 092, 2017

    2017

  42. [50]

    Universality for Random Tensors,

    R. Gurau, “Universality for Random Tensors,”Ann. Inst. H. Poincar´ e Proba. Stat., vol. 50, no. 4, pp. 1474–1525, 2014

    2014

  43. [51]

    Regular colored graphs of positive degree,

    R. Gurau and G. Schaeffer, “Regular colored graphs of positive degree,”Ann. Inst. Henri Poincar´ e Comb. Phys. Interact., vol. 3, pp. 257–320, 2016

    2016

  44. [52]

    J. A. Ellis-Monaghan and I. Moffatt,Graphs on surfaces: dualities, polynomials, and knots, vol. 84. Springer, 2013

    2013

  45. [53]

    Profiles of Permutations,

    M. Lugo, “Profiles of Permutations,”Electron. J. Combin., vol. 16, p. R99, 2009. 15

    2009

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.