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

After a brief presentation of random matrices as a random surface QFT approach to 2D quantum gravity, we focus on two crucial mathematical physics results: the implementation of the large N limit (N being here the size of the matrix) and of the double-scaling mechanism for matrix models. It is worth emphasizing that, in the large N limit, it is the planar surfaces which dominate. In the third section of the chapter we introduce tensor models, seen as a natural generalization, in dimension higher than two, of matrix models. The last section of the chapter presents a potential generalisation of the Bollobás–Riordan polynomial for tensor graphs (which are the Feynman graphs of the perturbative expansion of QFT tensor models).

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Asymptotic Expansion of the Multi-Orientable Random Tensor Model

The Electronic Journal of Combinatorics ◽

10.37236/4629 ◽

2015 ◽

Vol 22 (1) ◽

Cited By ~ 2

Author(s):

Eric Fusy ◽

Adrian Tanasa

Keyword(s):

Asymptotic Expansion ◽

Matrix Models ◽

Three Dimensional ◽

Natural Generalization ◽

General Term ◽

Tensor Model ◽

Half Integer ◽

Tensor Models ◽

Random Tensor

Three-dimensional random tensor models are a natural generalization of the celebrated matrix models. The associated tensor graphs, or 3D maps, can be classified with respect to a particular integer or half-integer, the degree of the respective graph. In this paper we analyze the general term of the asymptotic expanion in $N$, the size of the tensor, of a particular random tensor model, the multi-orientable tensor model. We perform their enumeration and we establish which are the dominant configurations of a given degree.

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The Sachdev–Ye–Kitaev (SYK) holographic model

Combinatorial Physics ◽

10.1093/oso/9780192895493.003.0015 ◽

2021 ◽

pp. 260-290

Author(s):

Adrian Tanasa

Keyword(s):

Gaussian Distribution ◽

Mechanical Model ◽

Quantum Mechanical ◽

Tensor Model ◽

Holographic Model ◽

Leading Order ◽

Tensor Models ◽

Feynman Graphs ◽

Non Gaussian ◽

Random Tensor

In this chapter, we first review the Sachdev–Ye–Kitaev (SYK) model, which is a quantum mechanical model of N fermions. The model is a quenched model, which means that the coupling constant is a random tensor with Gaussian distribution. The SYK model is dominated in the large N limit by melonic graphs, in the same way the tensor models presented in the previous three chapters are dominated by melonic graphs. We then present a purely graph theoretical proof of the melonic dominance of the SYK model. It is this property which led E. Witten to relate the SYK model to the coloured tensor model. In the rest of the chapter we deal with the so-called coloured SYK model, which is a particular case of the generalisation of the SYK model introduced by D. Gross and V. Rosenhaus. We first analyse in detail the leading order and next-to-leading order vacuum, two- and four-point Feynman graphs of this model. We then exhibit a thorough asymptotic combinatorial analysis of the Feynman graphs at an arbitrary order in the large N expansion. We end the chapter by an analysis of the effect of non-Gaussian distribution for the coupling of the model.

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MATRIX MODELS AND INTEGRABLE C<1 OPEN STRING THEORIES

International Journal of Modern Physics D ◽

10.1142/s0218271894000290 ◽

1994 ◽

Vol 03 (01) ◽

pp. 203-206

Author(s):

LAURENT HOUART

Keyword(s):

Ising Model ◽

Boundary Conditions ◽

Matrix Models ◽

Matrix Model ◽

Scaling Limit ◽

Open String ◽

String Equation ◽

Random Surfaces ◽

Ising Spins ◽

Double Scaling Limit

We study in the double scaling limit the two-matrix model which represents the sum over closed and open random surfaces coupled to an Ising model. The boundary conditions are characterized by the fact that the Ising spins sitting at the vertices of the boundaries are all in the same state. We obtain the string equation.

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Doubling of equations and universality in matrix models of random surfaces

Physics Letters B ◽

10.1016/0370-2693(90)90910-x ◽

1990 ◽

Vol 247 (2-3) ◽

pp. 363-369 ◽

Cited By ~ 18

Author(s):

C. Bachas ◽

P.M.S. Petropoulos

Keyword(s):

Matrix Models ◽

Random Surfaces

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TENSOR MODELS AND HIERARCHY OF n-ARY ALGEBRAS

International Journal of Modern Physics A ◽

10.1142/s0217751x1105381x ◽

2011 ◽

Vol 26 (19) ◽

pp. 3249-3258 ◽

Cited By ~ 12

Author(s):

NAOKI SASAKURA

Keyword(s):

Models Of Quantum Gravity ◽

Quantum Gravity ◽

Matrix Models ◽

Algebraic Structure ◽

The Other ◽

Invariant Form ◽

Infinitesimal Symmetry ◽

Tensor Models ◽

Symmetry Transformations

Tensor models are generalization of matrix models, and are studied as models of quantum gravity. It is shown that the symmetry of the rank-three tensor models is generated by a hierarchy of n-ary algebras starting from the usual commutator, and the 3-ary algebra symmetry reported in the previous paper is just a single sector of the whole structure. The condition for the Leibnitz rules of the n-ary algebras is discussed from the perspective of the invariance of the underlying algebra under the n-ary transformations. It is shown that the n-ary transformations which keep the underlying algebraic structure invariant form closed finite n-ary Lie subalgebras. It is also shown that, in physical settings, the 3-ary transformation practically generates only local infinitesimal symmetry transformations, and the other more nonlocal infinitesimal symmetry transformations of the tensor models are generated by higher n-ary transformations.

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A RENORMALIZATION PROCEDURE FOR TENSOR MODELS AND SCALAR-TENSOR THEORIES OF GRAVITY

International Journal of Modern Physics A ◽

10.1142/s0217751x10050433 ◽

2010 ◽

Vol 25 (23) ◽

pp. 4475-4492 ◽

Cited By ~ 5

Author(s):

NAOKI SASAKURA

Keyword(s):

General Relativity ◽

Models Of Quantum Gravity ◽

Quantum Gravity ◽

Matrix Models ◽

Dynamical Variable ◽

Renormalization Procedure ◽

Emergent Phenomena ◽

Tensor Models ◽

Theories Of Gravity

Tensor models are more-index generalizations of the so-called matrix models, and provide models of quantum gravity with the idea that spaces and general relativity are emergent phenomena. In this paper, a renormalization procedure for the tensor models whose dynamical variable is a totally symmetric real three-tensor is discussed. It is proven that configurations with certain Gaussian forms are the attractors of the three-tensor under the renormalization procedure. Since these Gaussian configurations are parametrized by a scalar and a symmetric two-tensor, it is argued that, in general situations, the infrared dynamics of the tensor models should be described by scalar-tensor theories of gravity.

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Colored Triangulations of Arbitrary Dimensions are Stuffed Walsh Maps

The Electronic Journal of Combinatorics ◽

10.37236/5614 ◽

2017 ◽

Vol 24 (1) ◽

Cited By ~ 6

Author(s):

Valentin Bonzom ◽

Luca Lionni ◽

Vincent Rivasseau

Keyword(s):

Matrix Models ◽

Building Blocks ◽

Field Method ◽

Fixed Number ◽

Colored Graphs ◽

Intermediate Field ◽

Tensor Models ◽

Combinatorial Maps ◽

Arbitrary Dimensions ◽

Number Of Faces

Regular edge-colored graphs encode colored triangulations of pseudo-manifolds. Here we study families of edge-colored graphs built from a finite but arbitrary set of building blocks, which extend the notion of $p$-angulations to arbitrary dimensions. We prove the existence of a bijection between any such family and some colored combinatorial maps which we call stuffed Walsh maps. Those maps generalize Walsh's representation of hypermaps as bipartite maps, by replacing the vertices which correspond to hyperedges with non-properly-edge-colored maps. This shows the equivalence of tensor models with multi-trace, multi-matrix models by extending the intermediate field method perturbatively to any model. We further use the bijection to study the graphs which maximize the number of faces at fixed number of vertices and provide examples where the corresponding stuffed Walsh maps can be completely characterized.

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