Bootstraps to strings: solving random matrix models with positivite


A new approach to solving random matrix models directly in the large N limit is developed. First, a set of numerical values for some low-pt correlation functions is guessed. The large N loop equations are then used to generate values of higher-pt correlation functions based on this guess. Then one tests whether these higher-pt functions are consistent with positivity requirements, e.g., (tr M2k) ≥ 0. If not, the guessed values are systematically ruled out. In this way, one can constrain the correlation functions of random matrices to a tiny subregion which contains (and perhaps converges to) the true solution. This approach is tested on single and multi-matrix models and handily reproduces known solutions. It also produces strong results for multi-matrix models which are not believed to be solvable. A tantalizing possibility is that this method could be used to search for new critical points, or string worldsheet theories.

A preprint version of the article is available at ArXiv.


  1. [1]

    G. ’t Hooft, A Planar Diagram Theory for Strong Interactions, Nucl. Phys.B 72 (1974) 461 [INSPIRE].

  2. [2]

    G. ’t Hooft, A Two-Dimensional Model for Mesons, Nucl. Phys.B 75 (1974) 461 [INSPIRE].

  3. [3]

    E.P. Wigner, Characteristic vectors of bordered matrices with infinite dimensions I, in The Collected Works of Eugene Paul Wigner, pp. 524–540, Springer (1955).

  4. [4]

    E. Brézin, C. Itzykson, G. Parisi and J.-B. Zuber, Planar diagrams, in The Large N Expansion In Quantum Field Theory And Statistical Physics: From Spin Systems to 2-Dimensional Gravity, pp. 567–583, World Scientific (1993).

  5. [5]

    D. Bessis, C. Itzykson and J.B. Zuber, Quantum field theory techniques in graphical enumeration, Adv. Appl. Math.1 (1980) 109 [INSPIRE].

    MathSciNet  Article  Google Scholar 

  6. [6]

    M.R. Douglas and S.H. Shenker, Strings in Less Than One-Dimension, Nucl. Phys.B 335 (1990) 635 [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  7. [7]

    V.A. Kazakov, Solvable matrix models, 2000, hep-th/0003064 [INSPIRE].

  8. [8]

    B. Eynard, Formal matrix integrals and combinatorics of maps, in Random matrices, random processes and integrable systems, pp. 415–442, Springer (2011).

  9. [9]

    I.R. Klebanov, J.M. Maldacena and N. Seiberg, Unitary and complex matrix models as 1-D type 0 strings, Commun. Math. Phys.252 (2004) 275 [hep-th/0309168] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  10. [10]

    P. Di Francesco, P.H. Ginsparg and J. Zinn-Justin, 2-D Gravity and random matrices, Phys. Rept.254 (1995) 1 [hep-th/9306153] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  11. [11]

    P.D. Anderson and M. Kruczenski, Loop Equations and bootstrap methods in the lattice, Nucl. Phys.B 921 (2017) 702 [arXiv:1612.08140] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  12. [12]

    A. Jevicki and J.P. Rodrigues, Master Variables and Spectrum Equations in Large N Theories, Nucl. Phys.B 230 (1984) 317 [INSPIRE].

    ADS  Article  Google Scholar 

  13. [13]

    R. Curto and L. Fialkow, The truncated complex k-moment problem, Trans. Am. Math. Soc.352 (2000) 2825.

    MathSciNet  Article  Google Scholar 

  14. [14]

    S. Burgdorf and I. Klep, The truncated tracial moment problem, J. Operator Theor.68 (2012) 141.

  15. [15]

    C. Itzykson and J.B. Zuber, The Planar Approximation. 2., J. Math. Phys.21 (1980) 411 [INSPIRE].

  16. [16]

    N. Ishibashi, H. Kawai, Y. Kitazawa and A. Tsuchiya, A Large N reduced model as superstring, Nucl. Phys.B 498 (1997) 467 [hep-th/9612115] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  17. [17]

    I.R. Klebanov, String theory in two-dimensions, in Spring School on String Theory and Quantum Gravity (to be followed by Workshop), Trieste, Italy, 15–23 April 1991, pp. 30–101 (1991) [hep-th/9108019] [INSPIRE].

  18. [18]

    I.R. Klebanov, F. Popov and G. Tarnopolsky, TASI Lectures on Large N Tensor Models, PoS(TASI2017)004 (2018) [arXiv:1808.09434] [INSPIRE].

  19. [19]

    B. Eynard, T. Kimura and S. Ribault, Random matrices, arXiv:1510.04430 [INSPIRE].

  20. [20]

    G.M. Cicuta, L. Molinari and E. Montaldi, Large N Phase Transitions in Low Dimensions, Mod. Phys. Lett.A 1 (1986) 125 [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  21. [21]

    R. Dijkgraaf and E. Witten, Developments in Topological Gravity, Int. J. Mod. Phys.A 33 (2018) 1830029 [arXiv:1804.03275] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  22. [22]

    B. Simon, The classical moment problem as a self-adjoint finite difference operator, Adv. Math.137 (1998) 82.

    MathSciNet  Article  Google Scholar 

  23. [23]

    M. Staudacher, Combinatorial solution of the two matrix model, Phys. Lett.B 305 (1993) 332 [hep-th/9301038] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  24. [24]

    D.A. Johnston, Symmetric vertex models on planar random graphs, Phys. Lett.B 463 (1999) 9 [cond-mat/9812169] [INSPIRE].

Download references

Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited

Author information



Corresponding author

Correspondence to Henry W. Lin.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

ArXiv ePrint: 2002.08387

Electronic supplementary material


(NB 170 kb)

Rights and permissions

This article is published under an open access license. Please check the 'Copyright Information' section for details of this license and what re-use is permitted. If your intended use exceeds what is permitted by the license or if you are unable to locate the licence and re-use information, please contact the Rights and Permissions team.

About this article

Verify currency and authenticity via CrossMark

Cite this article

Lin, H.W. Bootstraps to strings: solving random matrix models with positivite. J. High Energ. Phys. 2020, 90 (2020).

Download citation


  • Matrix Models
  • 1/N Expansion
  • Field Theories in Lower Dimensions