**Virtual Tensor Journal Club** === Time: on Wednesdays, at 2pm. Zoom link: https://cnrs.zoom.us/j/93766891861?pwd=emExT096ZlpwM1U0a3BHeS93ZEVxQT09 Meeting ID: 937 6689 1861 Password: emExT096ZlpwM1U0a3BHeS93ZEVxQT09 Past recordings: http://math.univ-lyon1.fr/homes-www/vignes/TJC/Archives Indico calendar: https://indico.math.cnrs.fr/category/415/ Current organizers: Sylvain Carrozza (sylvain.carrozza [at] u-bourgogne.fr), Luca Lionni (luca.lionni [at] ens-lyon.fr) and Reiko Toriumi (reiko.toriumi [at] oist.jp) ----------------------------------------------------------------------------- **Academic year 2024-2025** * May 14: **Liza Rozenberg**, Harvard * May 7: **Julian Sonner**, Université de Genève **Tensor models for chaotic CFT and 3D gravity** *Black holes are among the most extreme and mysterious objects known in physics. In this talk I will focus on their quantum chaotic nature, motivating the study of chaotic conformal field theories capable of describing the holographic dual of this behavior. These chaotic conformal field theories are naturally formulated in terms of mixed matrix-tensor models, which look tantalisingly closely related to tensor models of 3D gravity based on Virasoro symmetry.* * March 26: **Simon Lin**, NYU **Multipartite entanglement structure in random tensor networks and gravity** *I will talk about the classification of multi-partite entanglement in random tensor networks (RTN) via multipartite contractions of density matrices. In particular I will focus on two such quantities: the reflected entropy/Markov gap, and the multi-entropy. I will demonstrate how the calculation of such quantities in RTN naturally corresponds to a combinatorial optimization problem, and thus allowing explicit calculation in the large $N$ limit. This in turn allows for interesting connections to minimal cut problems on the graph defined by the RTN via max-flow/min-cut theorem. Implications and applications on quantum gravity and holography will also be discussed.* * March 5: **Zois Gyftopoulos**, Universität Heidelberg **Asymptotic safety meets tensor field theory: towards a new class of gravity-matter systems** *In this talk, I will combine asymptotically safe quantum gravity and a tensor field theory to exhibit the first example of a theory with gravity and scalar fields in four dimensions which may realize asymptotic safety at a non-vanishing value of the scalar quartic coupling. I will first present (further) evidence that in the asymptotic-safety paradigm, quantum fluctuations of gravity generically screen the quartic couplings in (multi-)scalar models. For a tensor field theory in which the scalar field transforms under an internal O(N)^3 symmetry, this has the effect of replacing asymptotic freedom, recently discovered at large N on a fixed flat background, by an interacting fixed point in the presence of quantum gravity.* * February 19: **Sanjaye Ramgoolam**, QMU London. <span style="color:red"> **Unusual time: 3pm (Paris)** </span> **Negative specific heat capacities from gauged matrix and tensor quantum harmonic oscillators** *Quantum harmonic oscillator systems with gauged large N symmetries have been shown to have thermodynamic properties similar to expectations from the quantum physics of small black holes in Anti-de-Sitter space. Explicit partition functions for matrix harmonic oscillators for matrix size N with S_N gauge symmetry are shown to give rise to negative specific heat capacities in the micro-canonical ensemble for energies below a bound scaling as ( N Log N ) for large N. In the canonical ensemble there is a sharp Hagedorn-like transition at a temperature, derivable from a high temperature expansion, which approaches zero in the large N limit. There is evidence, based on representation-theoretic number sequences for the counting of states, that similar thermodynamic features hold in multi-matrix and tensor systems with U(N) gauge symmetry. The talk is based on https://link.springer.com/article/10.1007/JHEP12(2024)161* * December 11: **Michael Winer**, IAS. <span style="color:red"> **Unusual time: 3pm (Paris)** </span> **Catastrophic Failure of the Large-N Expansion in a Bosonic Tensor Model** *We study the tensor model generalization of the quantum p-spherical model in the large-N limit. While the tensor model has the same large-N expansion as the disordered quantum p-spherical model, its ground state is superextensive, in contradiction with large-N perturbation theory. Therefore, the large-N expansion of this model catastrophically fails at arbitrarily large-N, without any obvious signal in perturbation theory.* * November 27: **Ludovic Fraser-Taliente**, University of Oxford **Free energy extremization in melonic conformal field theories** *Given enough supersymmetry, the IR limits of certain quantum field theories can be solved by extremizing a quantity associated to the sphere free energy $F=-\log Z_{S^d}$ over a space of trial data, subject to a constraint. This constraint effectively enforces the marginality of the interaction term in the IR. The tensorial quantum field theories admit a large-$N$ limit dominated by the melonic graphs; we show that their IR limits, the melonic CFTs, are determined by an identical constrained extremization principle. $F$ has been suggested as a counting of the number of degrees of freedom of a field theory, so we can understand this as: "as much 'stuff' as possible, given a constraint."* * November 13: **Vanessa Piccolo**, ENS Lyon **Dynamics of optimization in high dimensions for multi-spiked tensor PCA** *In this talk, we address the high-dimensional noisy tensor estimation problem known as multi-spiked tensor PCA (Principal Component Analysis). This problem involves estimating a finite number of unknown, orthogonal signal vectors (referred to as spikes) from noisy tensor observations. I will present new results, obtained in collaboration with Gérard Ben Arous (NYU) and Cédric Gerbelot (NYU), on the sample complexity and running time required for stochastic gradient descent to efficiently recover all spikes from random initialization. Our results reveal a sequential recovery process, in which each spike is recovered one after the other, ultimately leading to either exact recovery of each spike or recovery of a permutation of the full set.* ----------------------------------------------------------------------------- **Academic year 2023-2024** * June 26: **Rémi Avohou**, OIST **Counting $UO$ -tensorial invariants** *Invariants under the group $U(N)^{\otimes r} \otimes O(N)^{\otimes q}$ are generated through the contraction of complex tensors, which have an order of $r+q$, also denoted $(r,q)$. These tensors undergo transformations according to $r$ fundamental representations of the unitary group $U(N)$ and $q$ fundamental representations of the orthogonal group $O(N)$. Consequently, $U(N)^{\otimes r} \otimes O(N)^{\otimes q}$ invariants serve as observables in tensor models, possessing a tensor field with an order of $(r,q)$. In this presentation, I will show the enumeration of these observables using group theoretic formulae, for tensor fields with any given order $(r,q)$. For a general order $(r,q)$, such enumeration can be viewed as the partition function for a topological quantum field theory (TQFT), where the symmetric group acts as the gauge group. The discussion will include the identification of the $2$-complex pertaining to the enumeration of the invariants, which consequently defines the TQFT, and establish a correspondence with counting associated with covers of various topologies.* * June 19: **Tim Kunisky**, Yale **The finite free cumulants of a tensor and their statistical applications** *Consider hypothesis testing random matrices: given a draw from one of two matrix distributions, an algorithm must try to deduce which distribution it came from. If these distributions are orthogonally invariant---unchanged by conjugation by an orthogonal matrix---then a natural class of test statistics is the polynomials of a matrix that are themselves invariant under orthogonal conjugation. As is well-known, those polynomials are generated by traces of matrix powers. Classical invariant theory describes a similar situation for tensors: a canonical collection of polynomials given by evaluating "tensor networks" generates all invariant polynomials. Moreover, recent investigations in theoretical computer science suggest that, for many invariant matrix- and tensor-valued statistical problems, some such polynomial statistic should perform optimally for a given computational budget. It is then natural to ask: for specific invariant statistical problems, what is the performance of algorithms computing low-degree invariant polynomials?* *I will present new results on a remarkable collection of polynomials of a matrix or tensor that allow this question to be addressed in several cases. On the one hand, these polynomials provide an approximately orthonormal basis of the space of invariant polynomials with respect to an invariant Gaussian measure, the natural "null hypothesis" for many testing problems. On the other hand, they obey an additive free convolution rule analogous to the free cumulants (more precisely, their non-asymptotic analogs in finite free cumulants) of matrix-valued free probability theory. I will show how these properties, taken together, allow us to understand the cost of testing for exceptional principal components and for latent geometric structure in a random tensor. Finally, I will discuss how these results might pave the way to a fuller extension of free probability from random matrices to random tensors.* *Based on joint work with Cris Moore and Alex Wein.* * June 5: **Rémi Bonnin**, ENS Ulm **Random tensors : universality of the Wigner-Gurau limit** *In this talk we will develop a combinatorial approach for studying moments of the resolvent trace for random tensors proposed by Razvan Gurau, giving the maximal contribution of the expectation of trace invariants (or moments) of a tensor. Our work is based on the study of hypergraphs and extends the combinatorial proof of moments convergence for Wigner's theorem. This also opens up paths for research akin to free probability for random tensors. Specifically, trace invariants form a complete basis of tensor invariants and constitute the moments of the resolvent trace. For arandom tensor with entries independent, centered, with the right variance and bounded moments, we will show the convergence of the expectation and bound the variance of the balanced single trace invariant. This implies the universality of the convergence of the associated measure towards the law obtained by Gurau in the Gaussian case, whose limiting moments are given by the Fuss-Catalan numbers. Additionally, in the Gaussian case, the limiting distribution of the $k$-times contracted $p$-order random tensor by a deterministic vector is always the Wigner-Gurau law at order $p-k$, dilated by $\sqrt{\binom{p-1}{k}}$.* * May 29: **Naoki Sasakura**, Yukawa Institute **Signed eigenvalue/vector distribution of complex random tensor** *Eigenvalue/vector distributions of random tensors can be rewritten as partition functions of quantum field theories, that allows systematic, widely applicable, and powerful analysis of the distributions. In particular singed distributions can be rewritten as four-fermi theories, which are in principle always exactly computable. Though signed distributions are different from genuine distributions, they are expected to be intimately related and coincident near their endpoints, which have applications to such as geometric measure of entanglement, the largest eigenvalue, and the best rank-one approximation. In this talk, we apply the method to the signed eigenvalue/vector distribution of complex random tensor, and obtain an exact compact expression. We then compute the endpoint of the distribution in the large dimension limit and also that in the large order limit. An open question is pointed out concerning the latter.* * May 15: **Stéphane Dartois**, CEA List **Injective norm of random tensors and geometric entanglement of random quantum states** *In this talk, I will present the results of a collaboration with Benjamin McKenna on the injective norm of large random Gaussian tensors and uniform random quantum states, and describe some of the context underlying this work. The injective norm is a natural generalization to tensors of the operator norm of a matrix and appears in multiple fields. In quantum information, the injective norm is one important measure of genuine multipartite entanglement of quantum states, known as geometric entanglement. In our recent preprint, we provide a high-probability upper bound on the injective norm of real and complex Gaussian random tensors, which corresponds to a lower bound on the geometric entanglement of random quantum states, and to a bound on the ground-state energy of a particular multispecies spherical spin glass model. Our result represents a first step towards solving an important question in quantum information that has been part of folklore.* * April 3: **Victor Nador**, IMPAN Krakow **From the Boulatov to Amit-Roginsky models: a link between tensorial and melonic field theories** *Most tensor field theories are known for admitting a large N expansion which is dominated by melonic graphs. This feature is shared with some vectori theories as in the SYK model, or the Amit-Roginsky (AR) model. In this talk, I will present how a generalized version of the AR model can be obtained as a perturbation around classical solutions of the Boulatov model, a tensorial field theory endowed with additional group data. After an overview of the main features of the Boulatov model and its equation of motions, I will give necessary conditions on the classical solution to recover an Amit-Roginsky-like model as a perturbation around this solution. This result exhibits how a vector model exhibiting a large N melonic limit can arise from a richer tensorial theory with similar features. This talk is based on a collaboration with A. Tanasa, D. Oriti, X. Pang and Y. Wang.* * March 20: **Nadia Flodgren**, Stockholm University **Classifying large N limits of multiscalar theories by algebra** *We initiate a new approach to perturbative corrections in QFTs with multiple scalars and show that one-loop RG flows can be described in terms of a commutative but non-associative algebra. Through the use of the algebra and simple scaling arguments we can identify useful large N scalings of the couplings and large N limits. The algebraic concepts of subalgebras and ideals are used to characterise the corrections. We demonstrate this method for example models in 4D with O(N) symmetry, as well as for a multi-scalar theory with M SU(N) adjoint scalars. Using our method we classify all large N limits of these algebras: the standard ‘t Hooft limit, a ‘multi-matrix’ limit, and an intermediate case with extra symmetry and no free parameter. The algebra identifies these limits without the need of diagrammatic or combinatorial analysis.* * March 6: **Jan Pawlowski**, University of Heidelberg **Confinement & chiral symmetry breaking from functional approaches** *Functional approaches such as Dyson-Schwinger equations and the functional renormalisation group offer a concise access to the physics underlying many non-perturbative phenomena as well as a means for reaching quantitative results. This access is achieved via the solution of coupled sets of diagrammatic equations for full gauge-fixed QCD correlation functions. For these tasks, the gauge fixing rather allows for a convenient reparameterisation rather than being a liability, and provides direct access to the key QCD phenomena of confinement and chiral symmetry breaking. The talk starts with a brief introduction to the computation of correlation functions in gauge-fixed QCD with Dyson-Schwinger equations and the functional renormalisation group, mostly concentrating on the latter. Specifically, we discuss the mechanism behind dynamical chiral symmetry breaking and show that its occurrence is directly related to the strong coupling exceeding a critical value. It is also shown that in covariant gauges confinement, or the occurrence of a physical mass gap in Yang-Mills theory or QCD, is directly related to a mass gap in the gluonic propagator.* * February 21: **Juan Abranches**, OIST **Dually weighted multi-matrix models as a path to causal gravity-matter systems** *In this talk, I introduce a dually-weighted matrix model that reproduces two-dimensional Causal Dynamical Triangulations (CDT) coupled with the Ising Model. I present exact as well as approximate results for the Gaussian averages of characters of a Hermitian matrix A and A^2 for a given representation and show the present limitations that prevent us to solve the model analytically. This sets the stage for the formulation of more sophisticated matter models coupled to two-dimensional CDT as dually weighted multi-matrix models providing a complementary view to the standard simplicial formulation of CDT-matter models.* Fall semester 2023: * December 20: **Sari Ghanem**, University of Lübeck **Stability of Minkowski space-time governed by the Einstein-Yang-Mills system** *I shall start by presenting the Einstein-Yang-Mills equations and by writing them in the Lorenz gauge and in wave coordinates as a coupled system of non-linear hyperbolic partial differential equations, and I will then show how one constructs the initial data for a Cauchy hyperbolic formulation of the problem. Thereafter, I will present the philosophy behind the proof of the non-linear stability of the Minkowski space-time, solution to the Einstein-Yang-Mills system, in the Lorenz gauge and in wave coordinates, in all space dimensions greater or equal to three, based on a continuity argument for a higher order weighted energy norm. In the critical case of three space-dimensions, we use a null frame decomposition, that was first used by Lindblad and Rodnianski for the Einstein vacuum equations. We then deal with new difficulties that do not exist for Einstein vacuum nor for Einstein-Maxwell fields. In particular, we treat new terms that have a different structure in the non-linearities, and we derive a more refined formula to estimate the commutator term. This provides a new independent proof of the result by Mondal and Yau, that I posted on arXiv two months ago in a series of three papers that build up on each other, which cover all space dimensions greater or equal to three: Paper 1: Stability for space dimensions greater or equal to five (166 pages): https://arxiv.org/pdf/2310.07954.pdf , Paper 2: Exterior stability for four space dimensions (55 pages): https://arxiv.org/pdf/2310.08611.pdf , Paper 3: Exterior stability for three space dimensions (309 pages): https://arxiv.org/pdf/2310.08196.pdf .* * December 6: **Christian Jepsen**, KIAS **Noncritical and Critical O(N)r Models** *This talk will discuss the RG flows and fixed points of tensorial quantum field theories with global O(N)r symmetry, from the point of view of large N and small epsilon expansions. We will consider in turn theories with quartic and sextic interactions. For the quartic theories, we will find that the fixed points in the epsilon expansion can be matched to simple large N solutions, all dominated by vectorlike operators, whose leading Feynman diagrams are simple bubble diagrams, so that large N anomalous dimensions can be readily obtained via Hubbard-Stratonovich transforms. At general tensor rank r, the flow of vectorlike operators becomes an invariant subspace of the full RG flow and can be explicitly diagonalized, so that their fixed points can be enumerated. For the sextic theories, the large N dynamics differ qualitatively across different values of r, and we will see the appearance of a numerous and diverse set of perturbative fixed points starting at rank four.* * November 22: **Alicia Castro**, LaBRI, Université de Bordeaux **Scale-invariant random geometries from mating of trees** *In this talk, I present new results on the search for scale-invariant random geometries in the context of Quantum Gravity. To uncover new universality classes of such geometries, we generalized the mating of trees approach, which encodes Liouville Quantum Gravity on the 2-sphere in terms of a correlated Brownian motion describing a pair of random trees. We extended this approach to higher-dimensional correlated Brownian motions, leading to a family of non-planar random graphs that belong to new universality classes of scale-invariant random geometries. We developed a numerical method to efficiently simulate these random graphs and explore their scaling limits through distance measurements, allowing us to estimate Hausdorff dimensions in the two- and three-dimensional settings.* * November 8: **Ion Nechita**, Université de Toulouse **The asymptotic limit of random tensor flattenings** *Given a random tensor with i.i.d. entries, we consider the collection of all its flattenings (reshaping into matrices). We study the joint limit of these matrices in the sense of free probability. Under mild assumptions, we determine their limiting ∗-distribution in simple terms. We show that the limit is an operator-valued circular family, over the algebra CS_k of permutations. We identify free subfamilies and present applications to quantum information theory. All notions from (operator valued) free probability will be introduced during the talk, and no prior knowledge of free probability nor quantum information theory will be assumed. This is joint work with Stéphane Dartois and Camille Male, https://arxiv.org/abs/2307.11439*  * October 18: **Thomas Muller**, LaBRI, Université de Bordeaux **Duality of orthogonal and symplectic tensor models continued** *The groups O(N) and Sp(N) are related by an analytic continuation to negative values of N, O(−N) \approx Sp(N). This duality has been studied for vector models, some random matrix ensembles as well as colored tensor models with quartic interactions. In this talk, we will generalise this duality to tensor models with polynomial interactions of arbitrary order and with symmetry given by irreducible representations of O(N) and Sp(N). Based on : Lett Math Phys 113, 83 (2023), arXiv:2304.03625 and arXiv:2307.01527 (accepted in Jphys. A)* * October 4: **Sabine Harribey**, Nordita **Boundaries and Interfaces with Localized Cubic Interactions in the O(N) Model** *In this talk, I will present a new approach to boundaries and interfaces in the O(N) model where we add certain localized cubic interactions. These operators are nearly marginal when the bulk is four dimensional and they explicitly break the O(N) symmetry of the bulk theory down to O(N-1). The one-loop beta functions of the cubic couplings are then modified by the quartic bulk interactions. I will present the RG analysis for both the interface and the boundary cases. In the interface case, for sufficiently large N, we find stable IR fixed points with purely imaginary cubic couplings. In the boundary case, we find real fixed points for all N.I will also review theories of M pairs of symplectic fermions and one real scalar. The beta functions for these theories are related to those in the O(N) model via the replacement of N by 1-2M.Based on arXiv:2307.00072* * September 6: **Ada Altieri**, Université de Paris **Assessing the complexity of large interacting ecosystems through the lens of random matrix theory** *Many complex systems in Nature — from metabolic networks to ecosystems — appear to be poised at the edge of stability, hence displaying enormous responses to external perturbations. This marginal stability condition is often the consequence of the complex underlying interaction network, which can induce large-scale collective dynamics, and therefore critical behaviors. I will introduce the large-dimensional version of the Generalized Random Lotka-Volterra model with demographic noise and immigration. Leveraging on techniques rooted in the theory of spin glasses and random matrices, I will unveil a very rich — eventually hierarchical — structure in the organization of the equilibria and relate the slowing down of correlation functions to glassy-like properties. Finally, I will discuss extensions to non-logistic growth functions allowing for the description of positive feedback mechanisms as well as for the detection of novel phases as a smoking-gun signature of criticality.*