Advertisement

Constructive Matrix Theory for Higher-Order Interaction

  • Thomas Krajewski
  • Vincent RivasseauEmail author
  • Vasily Sazonov
Article
  • 12 Downloads

Abstract

This paper provides an extension of the constructive loop vertex expansion to stable matrix models with interactions of arbitrarily high order. We introduce a new representation for such models, then perform a forest expansion on this representation. It allows us to prove that the perturbation series of the free energy for such models is analytic in a domain uniform in the sizeNof the matrix.

Mathematics Subject Classification

81T08 

Notes

Acknowledgements

The work of VS was supported by the FWF Austrian funding agency through the Schroedinger fellowship J-3981.

Supplementary material

References

  1. 1.
    Rivasseau, V.: Constructive matrix theory. JHEP 0709, 008 (2007). arXiv:0706.1224 [hep-th]ADSMathSciNetCrossRefGoogle Scholar
  2. 2.
    Brydges, D., Kennedy, T.: Mayer expansions and the Hamilton–Jacobi equation. J. Stat. Phys. 48, 19 (1987)ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    Abdesselam, A., Rivasseau, V.: Trees, forests and jungles: a botanical garden for cluster expansions. In: Lecture Notes in Physics, vol 446. Springer, New York. arXiv:hep-th/9409094
  4. 4.
    Rivasseau, V., Wang, Z.: How to Resum Feynman graphs. Annales Henri Poincaré 15(11), 2069 (2014). arXiv:1304.5913 [math-ph]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Gurau, R., Ryan, J.P.: Colored tensor models—a review. SIGMA 8, 020 (2012). arXiv:1109.4812 [hep-th]MathSciNetzbMATHGoogle Scholar
  6. 6.
    Gurau, R.: Random Tensors. Oxford University Press, Oxford (2016)CrossRefzbMATHGoogle Scholar
  7. 7.
    ’t Hooft, G.: A planar diagram theory for strong interactions. Nucl. Phys. B 72, 461 (1974)ADSCrossRefGoogle Scholar
  8. 8.
    Gurau, R.: The 1/N expansion of colored tensor models. Annales Henri Poincaré 12, 829 (2011). arXiv:1011.2726 [gr-qc]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    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]ADSCrossRefGoogle Scholar
  10. 10.
    Gurau, R.: The complete 1/N expansion of colored tensor models in arbitrary dimension. Annales Henri Poincaré 13, 399 (2012). arXiv:1102.5759 [gr-qc]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Gurau, R., Krajewski, T.: Analyticity results for the cumulants in a random matrix model. Ann. Inst. Henri Poincaré D 2, 169–228 (2015). arXiv:1409.1705 [math-ph]MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Gurau, R.: The 1/N Expansion of Tensor Models Beyond Perturbation Theory. Commun. Math. Phys. 330, 973 (2014). arXiv:1304.2666 [math-ph]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Delepouve, T., Gurau, R., Rivasseau, V.: Universality and Borel summability of arbitrary quartic tensor models. Ann. Inst. Henri Poincaré Prob. Stat. 52, 821–848 (2016). arXiv:1403.0170 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Magnen, J., Rivasseau, V.: Constructive \(\phi ^4\) field theory without tears. Annales Henri Poincaré 9, 403 (2008). arXiv:0706.2457 [math-ph]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Zhao, F.-J.: Inductive Approach to Loop Vertex Expansion. arXiv:1809.01615
  16. 16.
    Gurau, R., Rivasseau, V.: The multiscale loop vertex expansion. Annales Henri Poincaré 16(8), 1869 (2015). arXiv:1312.7226 [math-ph]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Delepouve, T., Rivasseau, V.: Constructive tensor field theory: the \(T^4_3\) model. Commun. Math. Phys. 345, 477 (2016). arXiv:1412.5091 [math-ph]ADSCrossRefzbMATHGoogle Scholar
  18. 18.
    Lahoche, V.: Constructive tensorial group field Theory II: the \(U(1)-T^4_4\) model. J. Phys. A Math. Theor. arXiv:1510.05051 [hep-th]
  19. 19.
    Rivasseau, V., Vignes-Tourneret, F.: Constructive tensor field theory: The \(T^{4}_{4}\) model. Commun. Math. Phys. 366, 567 (2019). arXiv:1703.06510 [math-ph]ADSCrossRefzbMATHGoogle Scholar
  20. 20.
    Rivasseau, V.: Constructive tensor field theory. SIGMA 12, 085 (2016). arXiv:1603.07312 [math-ph]MathSciNetzbMATHGoogle Scholar
  21. 21.
    Rivasseau, V., Wang, Z.: Corrected loop vertex expansion for \(\Phi _2^4\) theory. J. Math. Phys. 56(6), 062301 (2015). arXiv:1406.7428 [math-ph]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    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]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Lionni, L., Rivasseau, V.: Note on the intermediate field representation of \(\phi ^{2k}\) theory in zero dimension. Math. Phys. Anal. Geom. 21(3), 23 (2018). arXiv:1601.02805 MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Lionni, L., Rivasseau, V.: Intermediate field representation for positive matrix and tensor interactions. To appear in Ann. Henri Poincaré.  https://doi.org/10.1007/s00023-019-00833-z, arXiv:1609.05018 [math-ph]
  25. 25.
    Rivasseau, V.: Loop vertex expansion for higher order interactions. Lett. Math. Phys. 108(5), 1147–1162 (2018). arXiv:1702.07602 [math-ph]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Gallavotti, G.: Perturbation Theory. In: Sen, R., Gersten, A. (eds.) Mathematical Physics Towards the XXI Century, pp. 275–294. Ben Gurion University Press, Ber Sheva (1994)Google Scholar
  27. 27.
    Abdesselam, A.: The Jacobian conjecture as a problem of perturbative quantum field theory. Annales Henri Poincaré 4, 199 (2003). arXiv:math/0208173 [math.CO]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    de Goursac, A., Sportiello, A., Tanasa, A.: The Jacobian conjecture, a reduction of the degree to the quadratic case. Annales Henri Poincaré 17(11), 3237 (2016). arXiv:1411.6558 [math.AG]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Abdesselam, A.: Feynman diagrams in algebraic combinatorics. Sém. Lothar. Combin. 49(2002/04), Art. B49c. arXiv:math/0212121
  30. 30.
    Eynard, B., Kimura, T., Ribault, S.: Random matrices. arXiv:1510.04430 [math-ph]
  31. 31.
    Krajewski, T., Rivasseau, V., Sazonov, V.: Work in preparation Google Scholar
  32. 32.
    Rivasseau, V.: Random tensors and quantum gravity. SIGMA 12, 069 (2016). arXiv:1603.07278 [math-ph]MathSciNetzbMATHGoogle Scholar
  33. 33.
    Mlotkowski, W., Penson, K.A.: “Probability distributions with binomial moments”, in infinite dimensional analysis. Quantum Prob. Relat. Top. 17(2), 1450014 (2014). World ScientificCrossRefzbMATHGoogle Scholar
  34. 34.
    Sokal, A.D.: An improvement of Watson’s theorem on Borel summability. J. Math. Phys. 21, 261 (1980)ADSMathSciNetCrossRefGoogle Scholar
  35. 35.
    Penson, K.A., Życzkowski, K.: Product of Ginibre matrices: Fuss–Catalan and Raney distributions. Phys. Rev. E 83, 061118 (2011)ADSCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Thomas Krajewski
    • 1
  • Vincent Rivasseau
    • 2
    Email author
  • Vasily Sazonov
    • 2
  1. 1.Centre de Physique Théorique, CNRS UMR 7332Université Aix-MarseilleMarseilleFrance
  2. 2.Laboratoire de Physique Théorique, CNRS UMR 8627Université Paris-SudOrsayFrance

Personalised recommendations