Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Universality and Borel summability of arbitrary quartic tensor models

Thibault Delepouve, Razvan Gurau, and Vincent Rivasseau

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We extend the study of melonic quartic tensor models to models with arbitrary quartic interactions. This extension requires a new version of the loop vertex expansion using several species of intermediate fields and iterated Cauchy–Schwarz inequalities. Borel summability is proven, uniformly as the tensor size $N$ becomes large. Every cumulant is written as a sum of explicitly calculated terms plus a remainder, suppressed in $1/N$. Together with the existence of the large $N$ limit of the second cumulant, this proves that the corresponding sequence of probability measures is uniformly bounded and obeys the tensorial universality theorem.


Nous étendons l’étude de modèles de tenseurs quartiques meloniques aux modèles avec des interactions quartiques arbitraires. Cette extension nécessite une nouvelle version du développement en vertex à boucles à l’aide de plusieurs nouveaux champs intermédiaires ainsi que l’utilisation répétée d’inégalités de Cauchy–Schwarz. La sommabilité de Borel est prouvée uniformément dans la taille $N$ du tenseur. Chaque cumulant est écrit comme une somme de termes explicitement calculés plus un reste supprimé à grand $N$. L’existence d’une limite finie à grand $N$ du second cumulant est établie et l’on démontre que la suite correspondante de mesures de probabilité est uniformément bornée en $N$ et obéit bien au théorème tensoriel d’universalité comme dans le cas melonique.

Article information

Ann. Inst. H. Poincaré Probab. Statist. Volume 52, Number 2 (2016), 821-848.

Received: 10 April 2014
Revised: 26 September 2014
Accepted: 8 October 2014
First available in Project Euclid: 4 May 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 81T08: Constructive quantum field theory

Random tensors Borel summability


Delepouve, Thibault; Gurau, Razvan; Rivasseau, Vincent. Universality and Borel summability of arbitrary quartic tensor models. Ann. Inst. H. Poincaré Probab. Statist. 52 (2016), no. 2, 821--848. doi:10.1214/14-AIHP655.

Export citation


  • [1] M. L. Mehta. Random Matrices, 3rd edition. Elsevier/Academic Press, Amsterdam, 2004.
  • [2] P. Di Francesco, P. H. Ginsparg and J. Zinn-Justin. 2-d gravity and random matrices. Phys. Rep. 254 (1995) 1–133. Available at hep-th/9306153.
  • [3] R. Gurau and J. P. Ryan. Colored tensor models – A review. SIGMA Symmetry Integrability Geom. Methods Appl. 8 020 (2012) 1–78. Available at arXiv:1109.4812 [hep-th].
  • [4] R. Gurau. Colored group field theory. Comm. Math. Phys. 304 (2011) 69–93. Available at arXiv:0907.2582 [hep-th].
  • [5] V. Bonzom, R. Gurau and V. Rivasseau. Random tensor models in the large $N$ limit: Uncoloring the colored tensor models. Phys. Rev. D 85 (2012) 084037. Available at arXiv:1202.3637 [hep-th].
  • [6] R. Gurau. The $1/N$ expansion of colored tensor models. Ann. Henri Poincaré 12 (2011) 829. Available at arXiv:1011.2726 [gr-qc].
  • [7] R. Gurau and V. Rivasseau. The $1/N$ expansion of colored tensor models in arbitrary dimension. Europhys. Lett. 95 (2011) 50004. Available at arXiv:1101.4182 [gr-qc].
  • [8] R. Gurau. The complete $1/N$ expansion of colored tensor models in arbitrary dimension. Ann. Henri Poincaré 13 (2012) 399–423. Available at arXiv:1102.5759 [gr-qc].
  • [9] V. Bonzom, R. Gurau, A. Riello and V. Rivasseau. Critical behavior of colored tensor models in the large $N$ limit. Nuclear Phys. B 853 (2011) 174–195. Available at arXiv:1105.3122 [hep-th].
  • [10] S. Dartois, R. Gurau and V. Rivasseau. Double scaling in tensor models with a quartic interaction. J. High Energy Phys. 9 088 (2013) 1–32. Available at arXiv:1307.5281 [hep-th].
  • [11] R. Gurau and G. Schaeffer. Regular colored graphs of positive degree. Available at arXiv:1307.5279 [math.CO].
  • [12] V. Bonzom. Multicritical tensor models and hard dimers on spherical random lattices. Phys. Lett. A 377 (2013) 501–506. Available at arXiv:1201.1931 [hep-th].
  • [13] V. Bonzom and H. Erbin. Coupling of hard dimers to dynamical lattices via random tensors. J. Stat. Mech. Theory Exp. 9 P09009 (2012) 1–18. Available at arXiv:1204.3798 [cond-mat.stat-mech].
  • [14] V. Bonzom. Revisiting random tensor models at large $N$ via the Schwinger–Dyson equations. J. High Energy Phys. 3 160 (2013) 1–24. Available at arXiv:1208.6216 [hep-th].
  • [15] V. Bonzom. New $1/N$ expansions in random tensor models. J. High Energy Phys. 6 062 (2013) 1–24. Available at arXiv:1211.1657 [hep-th].
  • [16] V. Bonzom and F. Combes. Fully packed loops on random surfaces and the $1/N$ expansion of tensor models. Available at arXiv:1304.4152 [hep-th].
  • [17] A. Baratin, S. Carrozza, D. Oriti, J. P. Ryan and M. Smerlak. Melonic phase transition in group field theory. Lett. Math. Phys. 104 (2014) 1003–1017. Available at arXiv:1307.5026 [hep-th].
  • [18] V. Rivasseau. The tensor track, III. Fortschr. Phys. 62 (2014) 81–107. Available at arXiv:1311.1461 [hep-th].
  • [19] R. Gurau. Universality for random tensors. Ann. Inst. Henri Poincaré Probab. Stat. 50 (2014) 1474–1525. Available at arXiv:1111.0519 [math.PR].
  • [20] R. Gurau. The $1/N$ expansion of tensor models beyond perturbation theory. Commun. Math. Phys. 330 (2014) 973–1019. Available at arXiv:1304.2666 [math-ph].
  • [21] V. Rivasseau. Constructive matrix theory. J. High Energy Phys. 9 008 (2007) 1–13. Available at arXiv:0706.1224 [hep-th].
  • [22] V. Rivasseau and Z. Wang. Loop vertex expansion for $\Phi^{2k}$ theory in zero dimension. J. Math. Phys. 51 092304 (2010) 1–17. Available at arXiv:1003.1037 [math-ph].
  • [23] J. Magnen and V. Rivasseau. Constructive $\phi^{4}$ field theory without tears. Ann. Henri Poincaré 9 (2008) 403–424. Available at arXiv:0706.2457 [math-ph].
  • [24] D. Brydges and T. Kennedy. Mayer expansions and the Hamilton–Jacobi equation. J. Stat. Phys. 48 (1987) 19–49.
  • [25] A. Abdesselam and V. Rivasseau. Trees, forests and jungles: A botanical garden for cluster expansions. In Constructive Physics, 7–36. V. Rivasseau (Ed.). Lecture Notes in Physics 446. Springer, Berlin, 1995. Available at arXiv:hep-th/9409094.
  • [26] B. Collins. Moments and cumulants of polynomial random variables on unitary groups, the Itzykson–Zuber integral, and free probability. Int. Math. Res. Not. IMRN 17 (2003) 953–982. Available at arXiv:math-ph/0205010.
  • [27] B. Collins and P. Sniady. Integration with respect to the Haar measure on unitary, orthogonal and symplectic group. Comm. Math. Phys. 264 (2006) 773–795. Available at arXiv:math-ph/0402073.
  • [28] D. Weingarten. Asymptotic behavior of group integrals in the limit of infinite rank. J. Math. Phys. 19 (1978) 999–1001.
  • [29] J. Magnen, K. Noui, V. Rivasseau and M. Smerlak. Scaling behaviour of three-dimensional group field theory. Classical Quantum Gravity 26 185012 (2009) 1–25. Available at arXiv:0906.5477 [hep-th].