# Constructive Matrix Theory for Higher-Order Interaction

Article

First Online:

- 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 size**N**of 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.Rivasseau, V.: Constructive matrix theory. JHEP
**0709**, 008 (2007). arXiv:0706.1224 [hep-th]ADSMathSciNetCrossRefGoogle Scholar - 2.Brydges, D., Kennedy, T.: Mayer expansions and the Hamilton–Jacobi equation. J. Stat. Phys.
**48**, 19 (1987)ADSMathSciNetCrossRefGoogle Scholar - 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.Rivasseau, V., Wang, Z.: How to Resum Feynman graphs. Annales Henri Poincaré
**15**(11), 2069 (2014). arXiv:1304.5913 [math-ph]ADSMathSciNetCrossRefzbMATHGoogle Scholar - 5.Gurau, R., Ryan, J.P.: Colored tensor models—a review. SIGMA
**8**, 020 (2012). arXiv:1109.4812 [hep-th]MathSciNetzbMATHGoogle Scholar - 6.Gurau, R.: Random Tensors. Oxford University Press, Oxford (2016)CrossRefzbMATHGoogle Scholar
- 7.’t Hooft, G.: A planar diagram theory for strong interactions. Nucl. Phys. B
**72**, 461 (1974)ADSCrossRefGoogle Scholar - 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.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.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.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.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.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.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.Zhao, F.-J.: Inductive Approach to Loop Vertex Expansion. arXiv:1809.01615
- 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.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.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.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.Rivasseau, V.: Constructive tensor field theory. SIGMA
**12**, 085 (2016). arXiv:1603.07312 [math-ph]MathSciNetzbMATHGoogle Scholar - 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.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.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.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.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.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.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.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.Abdesselam, A.: Feynman diagrams in algebraic combinatorics. Sém. Lothar. Combin.
**49**(2002/04), Art. B49c. arXiv:math/0212121 - 30.Eynard, B., Kimura, T., Ribault, S.: Random matrices. arXiv:1510.04430 [math-ph]
- 31.Krajewski, T., Rivasseau, V., Sazonov, V.: Work in preparation Google Scholar
- 32.Rivasseau, V.: Random tensors and quantum gravity. SIGMA
**12**, 069 (2016). arXiv:1603.07278 [math-ph]MathSciNetzbMATHGoogle Scholar - 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.Sokal, A.D.: An improvement of Watson’s theorem on Borel summability. J. Math. Phys.
**21**, 261 (1980)ADSMathSciNetCrossRefGoogle Scholar - 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