JCJC SIMI 2 - JCJC - SIMI 2 - Science informatique et applications

# Combinatorial physics, from matrix models to random tensor models – CombPhysMat2Tens

## Higher Dimensional Generalization of Combinatorial Physics Models

The main purpose of this project is to generalize in higher dimension (three and four) the different combinatorial developments of the celebrated (bi-dimensional) random matrix models.

### Combinatorial properties of random tensor models

Interactions between Combinatorics and Theoretical Physics have considerably increased in the last years. In this project, we propose to generalize to higher dimensions random matrix models, models widely studied both by combinatorists and by theoretical physicists. In parallel, as part of this project, we will use combinatorial techniques to achieve a better understanding of the Jacobian Conjecture, famous conjecture in mathematics.

The methods used are mainly combinatorial. Thus, we use algebraic combinatorics methods in order to give new (and more compact) proofs of the universality property of ribbon graph polynomials and matroids. Moreover, we use enumerative and analytic combinatorial methods to analyze generating functions of certain classes of three-dimensional maps. Finally, we use method of bijective combinatorics in order to reach a better understanding of objects with which we work with.

«Asymptotic expansion of the multi-orientable random tensor model», E. Fusy, A. Tanasa, arXiv:1408.5725, Electronic journal of combinatorics 22(1) (2015), #P1.52.
Random tensor models are associated in a canonical way with 3D maps, when the respective tensor have three indices. In this paper, we studied in detail the general term of the asymptotic expansion of N (N being the size of the tensor). In the 2D case, the corresponding development is implemented by the genus. In 3D, the role of the genus is played by the so-called degree of a 3D map. In this paper were determine what are the dominant configurations for each degree; generating functions for different classes of 3D maps were obtained and analyzed in detail.
«Analyticity results for the cumulants in a random matrix model», R. Gurau, T. Krajewski, arXiv:1409.1705, Annales IHP-D, Comb., Phys. Interactions 2 (2015) 169-228.
In general, in theoretical physics,the developments in Feynman graphs of perturbations of Gaussian integrals lead to non convergent series. This is the consequence of the rapid growth in the number of graphs as function of the graph's size. In this work, we develop a matrix integral on trees rather than on combinatorial maps. This gives rise to a convergent series, allowing to show that the cumulants are analytic functions and to control the rest when one gets back to the asymptotic expansion on combinatorial maps.
« The Jacobian Conjecture, a Reduction of the Degree to the Quadratic Case », A. de Goursac, A. Sportiello, A. Tanasa, Annales Henri Poincare 17 (2016) no.11, 3237-3254.
«O(N) Random Tensor Models » Sylvain Carrozza, Adrian Tanasa. Lett. Math. Phys. 106 (2016) no.11, 1531-1559.

In the rest of the project, we plan to continue to work on the themes initially planned. We intend, for example, to study various issues related to the continuum limit and to study several probabilistic aspects of these random tensor models. This could be achieved in collaboration with members of the combinatorics team of LABRI, with whom we recently linked several scientific contacts.

[1] «Asymptotic expansion of the multi-orientable random tensor model», E. Fusy, A. Tanasa, arXiv:1408.5725, Electronic journal of combinatorics 22(1) (2015), #P1.52.
[2] «Analyticity results for the cumulants in a random matrix model», R. Gurau, T. Krajewski, arXiv:1409.1705, Annales IHP-D, Comb., Phys. Interactions 2 (2015) 169-228.
[3] «The double scaling limit of random tensor models», V. Bonzom, R. Gurau, J. Ryan, A. Tanasa, arXiv:1404.7517, J. High Energy Phys. 1409 (2014) 051.
[4] «An analysis of the intermediate field theory of T^4 tensor model», V. Nguyen, S. Dartois, B. Eynard, arXiv:1409.5751, J. High Energy Phys. 1501 (2015) 013.
[5] «A Givental-like Formula and Bilinear Identities for Tensor Models », S. Dartois, arXiv:1409.5621, J. High Energy Phys. (à paraitre)
[6] «Polchinski's equation for group field theory», T. Krajewski, R. Toriumi, Fortsch. Phys. 62 (2014) 855-862.
[7] «Correlation functions of just renormalizable tensorial group field theory: the melonic approximation», D. Samary, C. Pérez-Sánchez, F. Vignes-Tourneret, R. Wulkenhaar, arXiv:1411.7213 [hep-th], Class. Quant. Grav. (à paraître)
[8] «An extension of the Bollobas-Riordan polynomial for vertex partitioned ribbon graphs: definition and universality», T. Krajewski, I. Moffat, A. Tanasa, Proceeding EuroComb 2015 (à paraître).
[9] «The double scaling limit of the multi-orientable random tensor model», R. Gurau, A. Tanasa, D. Youmans, Europhys. Lett. 111 (2015) 2, 21002.
[10] « The Jacobian Conjecture, a Reduction of the Degree to the Quadratic Case », A. de Goursac, A. Sportiello, A. Tanasa, Annales Henri Poincare 17 (2016) no.11, 3237-3254.
[11] «O(N) Random Tensor Models » Sylvain Carrozza, Adrian Tanasa. Lett. Math. Phys. 106 (2016) no.11, 1531-1559.

The interplay between combinatorics and theoretical physics has considerably increased over the last decades. This can be justified by the fact that a good knowledge of combinatorics allows theoretical physicists to propose better models, or to better formulate fundamental questions. On the other hand, mathematical physics techniques can be useful for proving various conjectures in combinatorics.
Our proposal gravitates around the combinatorial notion of ribbon graphs (or combinatorial maps, or graph embedded on surfaces) and their generalization to tensor graphs (or hyper-maps). We intend to study ribbon graphs in relation to the problem of colorability in graph theory, to establish a connection between Dixmier conjecture and quantum field theories based on these types of graphs. Finally, we intend to generalize ribbon graphs to tensor graphs and to study various combinatorial properties.
More concretely, a first task of our proposal is given by the study of the b-chromatic number problem for graphs embedded on surfaces. This is a completely original line of research and several interesting result scan be obtained (for example, the dependence of this b-chromatic number on the genus of the respective surface).
A second task is to analyze the Dixmier conjecture using combinatorial ribbon graph quantum field theoretical tools. Let us recall here that this conjecture (formulated in 1968) is maybe one of the most important open problems in mathematics. It simply states that any endomorphism of the Weyl algebra is an automorphism. We intend to look for a particular matrix model – seen as a quantum field theoretical model whose Feynman graphs are ribbon graphs. The combinatorics of such a (well chosen) model would allow for a reformulation of this conjecture.
A final task is given by the study of various combinatorial properties of random tensor models. Our first objective within this research line is a combinatorial algebraic one. We intend to implement a Connes-Kreimer algebra in order to describe the renormalization of such a tensor model. We then plan to investigate the large N (N being the size of the tensor) asymptotic expansion for some particular class of such tensor models. Moreover we intend to investigate the generalization of matrix integral techniques to tensor integral techniques. Let us emphasize here that this extremely ambitious last step in the programme could lead, if successful, to counting theorems for three- and four-dimensional maps, just as counting theorems for maps can be obtained via matrix integral techniques.
The second part of our project on the combinatorics of tensor graphs (which are dual to simplicial pseudo-manifolds in any dimensions) consists in generalizing the various existing bijections between certain classes of maps and labeled trees (Cori-Vauquelin-Schaeffer, Bouttier-Di Francesco-Guitter, Miermont, etc). Those bijections are indeed at the heart of the definition of the Brownian map. The construction of a random manifold in dimension three or four would be extremely interesting for physics in general, quantum gravity in particular and obviously for mathematics also.

These original objectives come as a continuation of the previous research results of the various members of our team. If successful, they will provide answers to difficult questions arising in graph theory, algebraic combinatorics or the combinatorics of various quantum field theoretical models, such as the random tensor models.

## Project coordination

Adrian Tanasa (Laboratoire Bordelais de Recherche Informatique)

The author of this summary is the project coordinator, who is responsible for the content of this summary. The ANR declines any responsibility as for its contents.

## Partner

LaBRI Laboratoire Bordelais de Recherche Informatique
LIPN Laboratoire d'Informatique Paris Nord

Help of the ANR 191,690 euros
Beginning and duration of the scientific project: March 2014 - 48 Months