Tensor models from the viewpoint of matrix models: the cases of loop models on random surfaces and of the Gaussian distribution

Bonzom, Valentin; Combes, Frédéric

doi:10.4171/aihpd/14

Cited by 12 publications

(16 citation statements)

References 63 publications

(168 reference statements)

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“…In our framework, the additional parameter D allows one to study the large D expansion of the planar diagrams. As highlighted in [20,21], this large D expansion is of fundamentally different nature depending on whether one uses the BGR scaling, as in [18], or the new enhanced scaling we propose, for the couplings τ a . With the BGR scaling, it is straightforward to show that the large N and the large D limits commute, whereas with the new scaling, it is essential to take N → ∞ first and D → ∞ second for the limit to make sense.…”

Section: Matrix-tensor Models and Applicationsmentioning

confidence: 98%

“…It was of course noticed that to the independent indices of tensor models can correspond spaces of different dimensions, so that tensor models in fact generalize the Wishart theory of random rectangular matrices. An interesting example, initiated in [18], consists in singling out two indices out of R = r + 2, (a 1 · · · a R ) = (abµ 1 · · · µ r ), and rewrite the tensor T in terms of a matrix X, with r additional tensor indices, T a 1 ···a R = (X µ 1 ···µr ) a b .…”

Section: Matrix-tensor Models and Applicationsmentioning

confidence: 99%

See 1 more Smart Citation

A New Large N Expansion for General Matrix–Tensor Models

Ferrari

Rivasseau

Valette

2019

Commun. Math. Phys.

102

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We define a new large N limit for general O(N ) R or U(N ) R invariant tensor models, based on an enhanced large N scaling of the coupling constants. The resulting large N expansion is organized in terms of a half-integer associated with Feynman graphs that we call the index. This index has a natural interpretation in terms of the many matrix models embedded in the tensor model. Our new scaling can be shown to be optimal for a wide class of non-melonic interactions, which includes all the maximally single-trace terms. Our construction allows to define a new large D expansion of the sum over diagrams of fixed genus in matrix models with an additional O(D) r global symmetry. When the interaction is the complete vertex of order R + 1, we identify in detail the leading order graphs for R a prime number. This slightly surprising condition is equivalent to the complete interaction being maximally singletrace.

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Section: Matrix-tensor Models and Applicationsmentioning

confidence: 98%

Section: Matrix-tensor Models and Applicationsmentioning

confidence: 99%

A New Large N Expansion for General Matrix–Tensor Models

Ferrari

Rivasseau

Valette

2019

Commun. Math. Phys.

102

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“…The latter is subdominant in the large-N limit, but the factor x 2(N 2 −N ) i is not, and must be taken into account. However, unlike the Vandermonde determinant, such term does not couple different eigenvalues, hence in the large-N limit we have a simple saddle point equation in which the eigenvalues are mutually decoupled: 16 For tensor models with a rectangular-matrix interpretation see also [60,61].…”

Section: A γ-Matrices In Odd-dimensionsmentioning

confidence: 99%

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

Benedetti

Delporte

2019

J. High Energ. Phys.

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Perturbing the standard Gross-Neveu model for N 3 fermions by quartic interactions with the appropriate tensorial contraction patterns, we reduce the original U (N 3 ) symmetry to either U (N ) × U (N 2 ) or U (N ) × U (N ) × U (N ). In the large-N limit, we show that in three dimensions such models admit new ultraviolet fixed points with reduced symmetry, besides the well-known one with maximal symmetry. The phase diagram notably presents a new phase with spontaneous symmetry breaking of one U (N ) component of the symmetry group. arXiv:1810.04583v2 [hep-th] 16 Nov 2018 1 Most commonly matrix models are presented as describing fields transforming in the adjoint representation, e.g. of U (N ). Such point of view is natural when introducing them as gauge fields, and wishing to discuss the different behavior of fields in the fundamental representation (flavor fields) and in the adjoint (connection fields) of the same group. Here we discuss them instead as fields in the fundamental representation of a smaller group, because we want to highlight the role of the smaller symmetry group for a given set of fields. Furthermore, the existence of a melonic large-N limit for tensors transforming in an irreducible representation of U (N ) (or O(N )) has been shown only very recently [7,8,9], and such models are less understood.2 Another way to obtain similar equations is to consider multi-matrix models at large N and in the limit of large number of matrices (see [10] and references therein).

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“…Since then, matrix models have been increasingly applied in many areas of mathematics and physics such as number theory, string theory, quantum gravity, holography, etc. Recently, matrix models have appeared to be intriguingly related to tensor theories [28][29][30]. However, the connection remains unclear so far.…”

Section: Introductionmentioning

confidence: 99%