The calculation of expectation values in Gaussian random tensor theory via meanders

Bonzom, Valentin; Combes, Frédéric

doi:10.4171/aihpd/13

Cited by 5 publications

(10 citation statements)

References 41 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For non-quartic bubbles, very few results exist, see for instance [6,7] where graphs with a single bubble, and for specific bubbles, are studied, focusing on those which maximize the number of faces. In a few instances, a non-direct approach was developed.…”

Section: Introductionmentioning

confidence: 99%

Colored Triangulations of Arbitrary Dimensions are Stuffed Walsh Maps

Bonzom

Lionni

Rivasseau

2017

Electron. J. Combin.

Self Cite

View full text Add to dashboard Cite

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. 1 The reason is in fact stronger than the bijection we describe. It is due to a non-perturbative equivalence which holds between tensor integrals and matrix integrals, which are respectively perturbative generating functions of colored graphs and colored maps. arXiv:1508.03805v1 [math.CO] 16 Aug 2015 2 One could use an arbitrary labeling of the black vertices and define a permutation τ Ω which maps the label of a white vertex to that of the black vertex it is paired with. Here we choose to set τ Ω = id.

show abstract

Section: Introductionmentioning

confidence: 99%

Colored Triangulations of Arbitrary Dimensions are Stuffed Walsh Maps

Bonzom

Lionni

Rivasseau

2017

Electron. J. Combin.

Self Cite

View full text Add to dashboard Cite

show abstract

“…For the sake of completeness, let us also mention that some relations between tensor models and matrix models have been investigated in [7]. Moreover, relations between tensor models and meanders have been studied in [6]. Finally, some relations between the counting of tensor model observables and branched covers of the two-sphere have been proved in [4].…”

Section: Introductionmentioning

confidence: 99%

The Multi-Orientable Random Tensor Model, a Review

Tanasă¹

2016

SIGMA

View full text Add to dashboard Cite

Abstract. After its introduction (initially within a group field theory framework) in [Tanasa A., J. Phys. A: Math. Theor. 45 (2012), 165401, 19 pages, arXiv:1109.0694], the multi-orientable (MO) tensor model grew over the last years into a solid alternative of the celebrated colored (and colored-like) random tensor model. In this paper we review the most important results of the study of this MO model: the implementation of the 1/N expansion and of the large N limit (N being the size of the tensor), the combinatorial analysis of the various terms of this expansion and finally, the recent implementation of a double scaling limit.

show abstract

“…The integral is performed using (6) and n denotes the number of matrix elements of U . Each term in the resulting sum over permutations σ, τ ∈ S n is represented as a diagram obtained from the one of the integrand by adding a dashed edge for each Kronecker delta.…”

Section: B Large N Behaviormentioning

confidence: 99%

“…The so-called melonic polynomials all have a single leading order Wick contraction (as well as a generalization of this family including nearly-melonic polynomials) [4]. A family of polynomials has been found in [6] whose numbers of leading order Wick contractions are numbers of meandric systems.…”

Section: Introductionmentioning

confidence: 99%

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

Bonzom¹,

Combes²

2015

Ann. Inst. Henri Poincaré Comb. Phys. Interact.

Self Cite

View full text Add to dashboard Cite

Observables in random tensor theory are polynomials in the entries of tensor of rank d which are invariant under U (N ) d . It is notoriously difficult to evaluate the expectations of such polynomials, even in the Gaussian distribution. In this article, we introduce singular value decompositions to evaluate the expectations of polynomial observables of Gaussian random tensors. Performing the matrix integrals over the unitary group leads to a notion of effective observables which expand onto regular, matrix trace invariants. Examples are given to illustrate that both asymptotic and exact new calculations of expectations can be performed this way.

show abstract

The calculation of expectation values in Gaussian random tensor theory via meanders

Cited by 5 publications

References 41 publications

Colored Triangulations of Arbitrary Dimensions are Stuffed Walsh Maps

Colored Triangulations of Arbitrary Dimensions are Stuffed Walsh Maps

The Multi-Orientable Random Tensor Model, a Review

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

Contact Info

Product

Resources

About