Communications in Mathematical Physics

, Volume 330, Issue 3, pp 973–1019 | Cite as

The 1/N Expansion of Tensor Models Beyond Perturbation Theory

  • Razvan GurauEmail author


We analyze in full mathematical rigor the most general quartically perturbed invariant probability measure for a random tensor. Using a version of the Loop Vertex Expansion (which we call the mixed expansion) we show that the cumulants write as explicit series in 1/N plus bounded rest terms. The mixed expansion recasts the problem of determining the subleading corrections in 1/N into a simple combinatorial problem of counting trees decorated by a finite number of loop edges.

As an aside, we use the mixed expansion to show that the (divergent) perturbative expansion of the tensor models is Borel summable and to prove that the cumulants respect an uniform scaling bound. In particular the quartically perturbed measures fall, in the N→ ∞ limit, in the universality class of Gaussian tensor models.


Matrix Model Tensor Model White Vertex External Edge Marked Vertex 
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  1. 1.
    Gurau, R., Ryan, J.P.: Colored tensor models—a review. SIGMA 8, 020 (2012). [arXiv:1109.4812 [hep-th]]
  2. 2.
    Mehta, M.L.: Random matrices. In: Pure and Applied Mathematics, Vol. 142, Amsterdam: Elsevier/Academic Press, 2004Google Scholar
  3. 3.
    Di Francesco, P., Ginsparg, P.H., Zinn-Justin, J.: 2-D Gravity and random matrices. Phys. Rept. 254, 1 (1995). [hep-th/9306153]Google Scholar
  4. 4.
    ’t Hooft G.: A planar diagram theory for strong interactions. Nucl. Phys. B 72, 461 (1974)ADSCrossRefMathSciNetGoogle Scholar
  5. 5.
    Brezin E., Itzykson C., Parisi G., Zuber J.B.: Planar diagrams. Commun. Math. Phys. 59, 35 (1978)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Kazakov V.A.: Bilocal regularization of models of random surfaces. Phys. Lett. B 150, 282 (1985)ADSCrossRefMathSciNetGoogle Scholar
  7. 7.
    David F.: A model of random surfaces with nontrivial critical behavior. Nucl. Phys. B 257, 543 (1985)ADSCrossRefGoogle Scholar
  8. 8.
    Oriti, D.: The microscopic dynamics of quantum space as a group field theory. arXiv:1110.5606 [hep-th]
  9. 9.
    Sasakura N.: Tensor model for gravity and orientability of manifold. Mod. Phys. Lett. A 6, 2613 (1991)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Ambjorn J., Durhuus B., Jonsson T.: Three-dimensional simplicial quantum gravity and generalized matrix models. Mod. Phys. Lett. A 6, 1133 (1991)ADSCrossRefMathSciNetGoogle Scholar
  11. 11.
    Sasakura, N.: Tensor models and 3-ary algebras. J. Math. Phys. 52, 103510 (2011). [arXiv:1104.1463 [hep-th]]Google Scholar
  12. 12.
    Sasakura, N.: Tensor models and hierarchy of n-ary algebras. Int. J. Mod. Phys. A 26, 3249 (2011). arXiv:1104.5312 [hep-th]
  13. 13.
    Boulatov, D.V.: A Model of three-dimensional lattice gravity. Mod. Phys. Lett. A 7, 1629 (1992). [hep-th/9202074]Google Scholar
  14. 14.
    Ooguri, H.: Topological lattice models in four-dimensions. Mod. Phys. Lett. A 7, 2799 (1992). [hep-th/9205090]Google Scholar
  15. 15.
    Baratin, A., Oriti, D.: Group field theory with non-commutative metric variables. Phys. Rev. Lett. 105 221302 (2010). [arXiv:1002.4723 [hep-th]]
  16. 16.
    Gurau, R.: Colored group field theory. Commun. Math. Phys. 304, 69. (2011). [arXiv:0907.2582 [hep-th]]
  17. 17.
    Gurau, R.: Lost in Translation: Topological Singularities in Group Field Theory. Class. Quant. Grav. 27, 235023 (2010). [arXiv:1006.0714 [hep-th]]
  18. 18.
    Bonzom, V., Gurau, R., Rivasseau, V.: Random tensor models in the large N limit: uncoloring the colored tensor models. Phys. Rev. D 85, 084037 (2012). [arXiv:1202.3637 [hep-th]]
  19. 19.
    Gurau, R.: The 1/N expansion of colored tensor models. Annales Henri Poincare 12, 829 (2011). [arXiv:1011.2726 [gr-qc]]
  20. 20.
    Gurau, R., Rivasseau, V.: The 1/N expansion of colored tensor models in arbitrary dimension. Europhys. Lett. 95, 50004 (2011). [arXiv:1101.4182 [gr-qc]]Google Scholar
  21. 21.
    Gurau, R.: The complete 1/N expansion of colored tensor models in arbitrary dimension. Annales Henri Poincare 13, 399 (2012). [arXiv:1102.5759 [gr-qc]]
  22. 22.
    Bonzom, V.: New 1/N expansions in random tensor models. J. High Energy phys. 2013, 62 (2013). arXiv:1211.1657 [hep-th]
  23. 23.
    Dartois, S., Rivasseau, V., Tanasa, A.: The 1/N expansion of multi-orientable random tensor models. Ann. Henri Poincare. doi: 10.1007/s00023-013-0262-8. arXiv:1301.1535 [hep-th]
  24. 24.
    Bonzom, V., Gurau, R., Riello, A., Rivasseau, V.: Critical behavior of colored tensor models in the large N limit. Nucl. Phys. B 853, 174 (2011). [arXiv:1105.3122 [hep-th]]
  25. 25.
    Gurau, R., Ryan, J.P.: Melons are branched polymers. Ann. Henri Poincare. doi: 10.1007/s00023-013-0291-3. arXiv:1302.4386 [math-ph]
  26. 26.
    Geloun, J.B., Magnen, J., Rivasseau, V.: Bosonic colored group field theory. Eur. Phys. J. C 70, 1119 (2010). arXiv:0911.1719 [hep-th]
  27. 27.
    Ryan, J.P.: Tensor models and embedded Riemann surfaces. Phys. Rev. D 85, 024010 (2012). [arXiv:1104.5471 [gr-qc]]
  28. 28.
    Carrozza, S., Oriti, D.: Bounding bubbles: the vertex representation of 3d Group Field Theory and the suppression of pseudo-manifolds. Phys. Rev. D 85, 044004 (2012). [arXiv:1104.5158 [hep-th
  29. 29.
    Carrozza, S., Oriti, D.: Bubbles and jackets: new scaling bounds in topological group field theories. JHEP 1206, 092 (2012). [arXiv:1203.5082 [hep-th]]
  30. 30.
    Bonzom, V., Gurau, R., Rivasseau, V.: The Ising Model on Random Lattices in Arbitrary Dimensions. arXiv:1108.6269 [hep-th]
  31. 31.
    Benedetti, D., Gurau, R.: Phase transition in dually weighted colored tensor models. Nucl. Phys. B 855, 420 (2012). arXiv:1108.5389 [hep-th]
  32. 32.
    Gurau, R.: The double scaling limit in arbitrary dimensions: a toy model. Phys. Rev. D 84, 124051 (2011). arXiv:1110.2460 [hep-th]
  33. 33.
    Gurau, R.: A generalization of the Virasoro algebra to arbitrary dimensions. Nucl. Phys. B 852, 592 (2011). [arXiv:1105.6072 [hep-th]]
  34. 34.
    Gurau, R.: The Schwinger Dyson equations and the algebra of constraints of random tensor models at all orders. Nucl. Phys. B 865, 133 (2012). [arXiv:1203.4965 [hep-th]]
  35. 35.
    Krajewski, T.: Schwinger-Dyson equations in group field theories of quantum gravity. arXiv:1211.1244 [math-ph]
  36. 36.
    Bonzom, V.: Revisiting random tensor models at large N via the Schwinger-Dyson equations. J. High Energy Phys. 2013, 160 (2013). arXiv:1208.6216 [hep-th]
  37. 37.
    Bonzom, V.:Multicritical tensor models and hard dimers on spherical random lattices. Phys. Lett. A 377(7), 501–506 (2013). arXiv:1201.1931 [hep-th]Google Scholar
  38. 38.
    Bonzom, V., Erbin, H.: Coupling of hard dimers to dynamical lattices via random tensors. J. Stat. Mech. (2012). P09009. arXiv:1204.3798 [cond-mat.stat-mech]
  39. 39.
    Ben Geloun, J., Rivasseau, V.: A renormalizable 4-dimensional tensor field theory. Commun. Math. Phys. 318(1), 69–109 (2013). arXiv:1111.4997 [hep-th]Google Scholar
  40. 40.
    Ben Geloun, J., Samary, D. O.: 3D tensor field theory: Renormalization and One-loop β-functions. arXiv:1201.0176 [hep-th]
  41. 41.
    Ben Geloun, J.: Two and four-loop β-functions of rank 4 renormalizable tensor field theories. Ann. Henri Poincare 14(6), 1599–1642 (2013). arXiv:1205.5513 [hep-th]
  42. 42.
    Geloun, J.B.: Asymptotic Freedom of Rank 4 Tensor Group Field Theory. arXiv:1210.5490 [hep-th]
  43. 43.
    Samary, D.O.: Beta functions of U(1)d gauge invariant just renormalizable tensor models. Phys. Rev. D. 88, 105003 (2013). arXiv:1303.7256 [hep-th]
  44. 44.
    Geloun, J.B., Livine, E.R.: Some classes of renormalizable tensor models. J. Math. Phys. 54, 082303 (2013). arXiv:1207.0416 [hep-th]Google Scholar
  45. 45.
    Carrozza, S., Oriti, D., Rivasseau, V.: Renormalization of tensorial group field theories: Abelian U(1) models in four dimensions. Commun. Math. Phys. (2014, to appear). arXiv:1207.6734 [hep-th]
  46. 46.
    Carrozza, S., Oriti, D., Rivasseau, V.: Renormalization of an SU(2) tensorial group field theory in three dimensions. Commun. Math. Phys. (2014, to appear). arXiv:1303.6772 [hep-th]
  47. 47.
    Rivasseau, V.:Quantum gravity and renormalization: the tensor track. AIP Conf. Proc. 1444, 18 (2012). arXiv:1112.5104 [hep-th]
  48. 48.
    Rivasseau, V.: The Tensor Track: an Update. arXiv:1209.5284 [hep-th]
  49. 49.
    Gurau, R.: Universality for Random Tensors. arXiv:1111.0519 [math.PR]
  50. 50.
    Glimm, J., Jaffe, A.: Quantum Physics. A functional integral point of view, 2nd edn. Berlin: Springer, 1987Google Scholar
  51. 51.
    Rivasseau, V.: Constructive matrix theory. JHEP 0709, 008 (2007). [arXiv:0706.1224 [hep-th]]
  52. 52.
    Rivasseau, V., Wang, Z.: Loop vertex expansion for phi**2K theory in zero dimension. J. Math. Phys. 51, 092304 (2010). [arXiv:1003.1037 [math-ph]]Google Scholar
  53. 53.
    Magnen, J., Noui, K., Rivasseau, V., Smerlak, M.: Scaling behaviour of three-dimensional group field theory. Class. Quant. Grav. 26, 185012 (2009). [arXiv:0906.5477 [hep-th]]Google Scholar
  54. 54.
    Collins, B.: Moments and cumulants of polynomial random variables on unitary groups, the Itzykson-Zuber integral, and free probability. Int. Math. Res. Not. 17, 953 (2003). [arXiv:math-ph/0205010]
  55. 55.
    Collins, B., Sniady, P.: Integration with respect to the Haar measure on unitary, orthogonal and symplectic group. Commun. Math. Phys. 264, 773 (2006). [arXiv:math-ph/0402073 ]
  56. 56.
    Pezzana M.: Sulla struttura topologica delle varietà compatte. Atti Sem. Mat. Fis. Univ. Modena 23, 269–277 (1974)Google Scholar
  57. 57.
    Ferri, M., Gagliardi, C.: Crystallisation moves. Pac. J. Math. 100(1), (1982)Google Scholar
  58. 58.
    Magnen J., Seneor R.: Phase space cell expansion and borel summability for the Euclidean phi**4 in three-dimensions theory. Commun. Math. Phys. 56, 237 (1977)ADSCrossRefMathSciNetGoogle Scholar
  59. 59.
    Feldman J., Magnen J., Rivasseau V., Seneor R.: Construction and Borel summability of infrared phi**4 in four-dimensions by a phase space expansion. Commun. Math. Phys. 109, 437 (1987)ADSCrossRefMathSciNetGoogle Scholar
  60. 60.
    Sokal A. D.: An improvement of Watson’s theorem on Borel summability. J. Math. Phys. 21, 261 (1980)ADSCrossRefMathSciNetGoogle Scholar
  61. 61.
    Rivasseau, V., Wang, Z.: How to Resum Feynman Graphs. Ann. Henri Poincare. doi: 10.1007/s00023-013-0299-8. arXiv:1304.5913 [math-ph]
  62. 62.
    Abdesselam, A., Rivasseau, V.: Trees, forests and jungles: a botanical garden for cluster expansions. In: Constructive physics, ed by V. Rivasseau. Lecture Notes in Physics, Vol. 446, Berlin: Springer, 1995Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.CPHT-UMR 7644, CNRSÉcole PolytechniquePalaiseau CedexFrance
  2. 2.Perimeter Institute for Theoretical PhysicsWaterlooCanada

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