Skip to main content

Advertisement

SpringerLink
Log in

Universality in several-matrix models via approximate transport maps

  • Published:
Acta Mathematica

Abstract

We construct approximate transport maps for perturbative several-matrix models. As a consequence, we deduce that local statistics have the same asymptotic as in the case of independent GUE or GOE matrices (i.e., they are given by the sine-kernel in the bulk and the Tracy–Widom distribution at the edge), and we show averaged energy universality (i.e., universality for averages of m-points correlation functions around some energy level E in the bulk). As a corollary, these results yield universality for self-adjoint polynomials in several independent GUE or GOE matrices which are close to the identity.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Akemann G., Ipsen J. R.: Recent exact and asymptotic results for products of independent random matrices. Acta Phys. Polon. B 46, 1747–1784 (2015)

    Article  MathSciNet  Google Scholar 

  2. Anderson, G. W., Guionnet, A. & Zeitouni, O., An Introduction to Random Matrices. Cambridge Studies in Advanced Mathematics, 118. Cambridge Univ. Press, Cambridge, 2010.

  3. Aptekarev A. I., Bleher P. M., Kuijlaars A. B. J.: Large n limit of Gaussian random matrices with external source. II. Comm. Math. Phys. 259, 367–389 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bekerman, F., Transport maps for \({\beta}\)-matrix models in the multi-cut regime. Preprint, 2015. arXiv:1512.00302 [math.PR].

  5. Bekerman F., Figalli A., Guionnet A.: Transport maps for \({\beta}\)-matrix models and universality. Comm. Math. Phys. 338, 589–619 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  6. Ben Arous G., Bourgade P.: Extreme gaps between eigenvalues of random matrices. Ann. Probab. 41, 2648–2681 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  7. Ben Arous G., Guionnet A.: Large deviations for Wigner’s law and Voiculescu’s non-commutative entropy. Probab. Theory Related Fields 108, 517–542 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bertola, M., Two-matrix models and biorthogonal polynomials, in The Oxford Handbook of Random Matrix Theory, pp. 310–328. Oxford Univ. Press, Oxford, 2011.

  9. Borot G., Guionnet A.: Asymptotic expansion of \({\beta}\) matrix models in the one-cut regime. Comm. Math. Phys. 317, 447–483 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  10. Borot, G. & Guionnet, A., Asymptotic expansion of beta matrix models in the multi-cut regime. Preprint, 2013. arXiv:1303.1045 [math.PR].

  11. Borot G., Guionnet A., Kozlowski K. K.: Large-N asymptotic expansion for mean field models with Coulomb gas interaction. Int. Math. Res. Not. IMRN 2015, 10451–10524 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  12. Bourgade, P., Erdős, L. & Yau, H.-T., Bulk universality of general \({\beta}\)-ensembles with non-convex potential. J. Math. Phys., 53 (2012), 095221, 19 pp.

  13. Bourgade P., Erdős L., Yau H.-T.: Edge universality of beta ensembles. Comm. Math. Phys. 332, 261–353 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  14. Bourgade P., Erdős L., Yau H.-T.: Universality of general \({\beta}\)-ensembles. Duke Math. J. 163, 1127–1190 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  15. Bourgade P., Erdős L., Yau H.-T., Yin J.: Fixed energy universality for generalized Wigner matrices. Comm. Pure Appl. Math. 69, 1815–1881 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  16. Brézin E., Itzykson C., Parisi G., Zuber J. B.: Planar diagrams. Comm. Math. Phys. 59, 35–51 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  17. Capitaine M., Péché S.: Fluctuations at the edges of the spectrum of the full rank deformed GUE. Probab. Theory Related Fields 165, 117–161 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  18. Collins B., Guionnet A., Maurel-Segala É.: Asymptotics of unitary and orthogonal matrix integrals. Adv. Math. 222, 172–215 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  19. Deift, P. A., Orthogonal Polynomials and Random Matrices: A Riemann–Hilbert Approach. Courant Lecture Notes in Mathematics, 3. Courant Institute of Mathematical Sciences, New York; Amer. Math. Soc., Providence, RI, 1999.

  20. Deift P. A., Gioev D.: Universality at the edge of the spectrum for unitary, orthogonal, and symplectic ensembles of random matrices. Comm. Pure Appl. Math. 60, 867–910 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  21. Deift, P. A. & Gioev, D., Universality in random matrix theory for orthogonal and symplectic ensembles. Int. Math. Res. Pap. IMRP, 2 (2007), Art. ID rpm004, 116 pp.

  22. Deift, P. A. & Gioev, D., Random Matrix Theory: Invariant Ensembles and Universality. Courant Lecture Notes in Mathematics, 18. Courant Institute of Mathematical Sciences, New York; Amer. Math. Soc., Providence, RI, 2009.

  23. Erdős, L., Universality of Wigner random matrices, in XVI International Congress on Mathematical Physics, pp. 86–105. World Sci., Hackensack, NJ, 2010.

  24. Erdős L., Knowles A., Yau H.-T., Yin J.: Delocalization and diffusion profile for random band matrices. Comm. Math. Phys. 323, 367–416 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  25. Erdős L., Péché S., Ramírez J. A., Schlein B., Yau H.-T.: Bulk universality for Wigner matrices. Comm. Pure Appl. Math. 63, 895–925 (2010)

    MathSciNet  MATH  Google Scholar 

  26. Erdős L., Schlein B., Yau H.-T., Yin J.: The local relaxation flow approach to universality of the local statistics for random matrices. Ann. Inst. Henri Poincaré Probab. Stat. 48, 1–46 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  27. Erdős L., Yau H.-T.: Universality of local spectral statistics of random matrices. Bull. Amer. Math. Soc. 49, 377–414 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  28. Erdős L., Yau H.-T.: Gap universality of generalized Wigner and \({\beta}\)-ensembles. J. Eur. Math. Soc. (JEMS) 17, 1927–2036 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  29. Erdős L., Yau H.-T., Yin J.: Rigidity of eigenvalues of generalized Wigner matrices. Adv. Math. 229, 1435–1515 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  30. Eynard B., Bonnet G.: The Potts-q random matrix model: loop equations, critical exponents, and rational case. Phys. Lett. B 463, 273–279 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  31. Forrester, P. J., Log-Gases and Random Matrices. London Mathematical Society Monographs Series, 34. Princeton Univ. Press, Princeton, NJ, 2010.

  32. Götze F., Venker M.: Local universality of repulsive particle systems and random matrices. Ann. Probab. 42, 2207–2242 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  33. Guionnet, A., Jones, V. F. R. & Shlyakhtenko, D., Random matrices, free probability, planar algebras and subfactors, in Quanta of Maths, Clay Math. Proc., 11, pp. 201–239. Amer. Math. Soc., Providence, RI, 2010.

  34. Guionnet A., Jones V. F. R., Shlyakhtenko D., Zinn-Justin P.: Loop models, random matrices and planar algebras. Comm. Math. Phys. 316, 45–97 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  35. Guionnet A., Maurel-Segala É.: Combinatorial aspects of matrix models. ALEA Lat. Am. J. Probab. Math. Stat. 1, 241–279 (2006)

    MathSciNet  MATH  Google Scholar 

  36. Guionnet A., Maurel-Segala É.: Second order asymptotics for matrix models. Ann. Probab. 35, 2160–2212 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  37. Guionnet A., Novak J.: Asymptotics of unitary multimatrix models: the Schwinger–Dyson lattice and topological recursion. J. Funct. Anal. 268, 851–2905 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  38. Guionnet A., Shlyakhtenko D.: Free monotone transport. Invent. Math. 197, 613–661 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  39. Haagerup U., Thorbjørnsen S.: A new application of random matrices: \({{\rm Ext}(C^*_{\rm red}(F_2))}\) is not a group. Ann. of Math. 162, 711–775 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  40. Kostov, I., Two-dimensional quantum gravity, in The Oxford Handbook of Random Matrix Theory, pp. 619–640. Oxford Univ. Press, Oxford, 2011.

  41. Kriecherbauer, T. & Shcherbina, M., Fluctuations of eigenvalues of matrix models and their applications. Preprint, 2010. arXiv:1003.6121 [math-ph].

  42. Kriecherbauer, T. & Venker, M., Edge statistics for a class of repulsive particle systems. Preprint, 2015. arXiv:1501.07501 [math.PR].

  43. Krishnapur M., Rider B., Virág B.: Universality of the stochastic Airy operator. Comm. Pure Appl. Math. 69, 145–199 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  44. Lee J. O., Schnelli K., Stetler B., Yau H.-T.: Bulk universality for deformed Wigner matrices. Ann. Probab. 44, 2349–2425 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  45. Levin E., Lubinsky D. S.: Universality limits in the bulk for varying measures. Adv. Math. 219, 743–779 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  46. Liu D. Z., Wang Y.: Universality for Products of Random Matrices I: Ginibre and Truncated Unitary Cases. Int. Math. Res. Not. IMRN 2016, 3473–3524 (2016)

    Article  MathSciNet  Google Scholar 

  47. Lubinsky, D. S., Universality limits via “old style” analysis, in Random Matrix Theory, Interacting Particle Systems, and Integrable Systems, Math. Sci. Res. Inst. Publ., 65, pp. 277–292. Cambridge Univ. Press, New York, 2014.

  48. Maïda M., Maurel-Segala É.: Free transport-entropy inequalities for non-convex potentials and application to concentration for random matrices. Probab. Theory Related Fields 159, 329–356 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  49. Male C.: The norm of polynomials in large random and deterministic matrices. Probab. Theory Related Fields 154, 477–532 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  50. Maurel-Segala, É., High order expansion of matrix models and enumeration of maps. Preprint, 2006. arXiv:math/0608192 [math.PR].

  51. Mehta M. L.: A method of integration over matrix variables. Comm. Math. Phys. 79, 327–340 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  52. Mehta, M. L., Random Matrices. Pure and Applied Mathematics, 142. Elsevier/Academic Press, Amsterdam, 2004.

  53. Nelson B.: Free transport for finite depth subfactor planar algebras. J. Funct. Anal. 268, 2586–2620 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  54. Ramírez J. A., Rider B., Virág B.: Beta ensembles, stochastic Airy spectrum, and a diffusion. J. Amer. Math. Soc. 24, 919–944 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  55. Shcherbina M.: On universality for orthogonal ensembles of random matrices. Comm. Math. Phys. 285, 957–974 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  56. Shcherbina, M., Change of variables as a method to study general \({\beta}\)-models: bulk universality. J. Math. Phys., 55 (2014), 043504, 23 pp.

  57. Shcherbina T.: Universality of the local regime for the block band matrices with a finite number of blocks. J. Stat. Phys. 155, 466–499 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  58. Tao T.: The asymptotic distribution of a single eigenvalue gap of a Wigner matrix. Probab. Theory Related Fields 157, 81–106 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  59. Tao T., Vu V.: Random matrices: universality of ESDs and the circular law. Ann. Probab. 38, 2023–2065 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  60. Tao T., Vu V.: Random matrices: universality of local eigenvalue statistics. Acta Math. 206, 127–204 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  61. Tao T., Vu V.: Random covariance matrices: universality of local statistics of eigenvalues. Ann. Probab. 40, 1285–1315 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  62. Tao T., Vu V.: Random matrices: universality of local spectral statistics of non-Hermitian matrices. Ann. Probab. 43, 782–874 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  63. Tracy C. A., Widom H.: Level-spacing distributions and the Airy kernel. Comm. Math. Phys. 159, 151–174 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  64. Tracy C. A., Widom H.: Level spacing distributions and the Bessel kernel. Comm. Math. Phys. 161, 289–309 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  65. Valkó B., Virág B.: Continuum limits of random matrices and the Brownian carousel. Invent. Math. 177, 463–508 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  66. Venker M.: Particle systems with repulsion exponent \({\beta}\) and random matrices. Electron. Commun. Probab. 18(83), 12 (2013)

    MathSciNet  MATH  Google Scholar 

  67. Voiculescu D.: Limit laws for random matrices and free products. Invent. Math. 104, 201–220 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  68. Wigner E. P.: Characteristic vectors of bordered matrices with infinite dimensions. Ann. of Math. 62, 548–564 (1955)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, ETH Zürich, Rämistrasse 101, CH-8092, Zürich, Switzerland

    Alessio Figalli

  2. Université de Lyon, École Normale Supérieure de Lyon, site Monod, UMPA UMR 5669 CNRS, 46, allée d’Italie, FR-69364, Lyon Cedex 07, France

    Alice Guionnet

Authors
  1. Alessio Figalli
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alice Guionnet
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alice Guionnet.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Figalli, A., Guionnet, A. Universality in several-matrix models via approximate transport maps. Acta Math 217, 81–176 (2016). https://doi.org/10.1007/s11511-016-0142-4

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11511-016-0142-4

Search

Navigation