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Free analysis and random matrices

  • Takagi Lectures
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Japanese Journal of Mathematics Aims and scope

Abstract

We describe the Schwinger–Dyson equation related with the free difference quotient. Such an equation appears in different fields such as combinatorics (via the problem of the enumeration of planar maps), operator algebra (via the definition of a natural integration by parts in free probability), in classical probability (via random matrices or particles in repulsive interaction). In these lecture notes, we shall discuss when this equation uniquely defines the system and in such a case how it leads to deep properties of the solution. This analysis can be extended to systems which approximately satisfy these equations, such as random matrices or Coulomb gas interacting particle systems.

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References

  1. G.W. Anderson, A. Guionnet and O. Zeitouni, An Introduction to Random Matrices, Cambridge Stud. Adv. Math., 118, Cambridge Univ. Press, Cambridge, 2010.

  2. 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 

  3. F. Bekerman, Transport maps for \({\beta}\)-matrix models in the multi-cut regime, preprint, arXiv:1512.00302.

  4. Biane P., Speicher R.: Stochastic calculus with respect to free Brownian motion and analysis on Wigner space. Probab. Theory Related Fields, 112, 373–409 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  5. Biane P., Speicher R.: Free diffusions, free entropy and free Fisher information. Ann. Inst. H. Poincaré Probab. Statist., 37, 581–606 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  6. A. Borodin, V. Gorin and A. Guionnet, Gaussian asymptotics of discrete \({\beta}\)-ensembles, preprint, arXiv:1505.03760.

  7. G. Borot and B. Eynard, All order asymptotics of hyperbolic knot invariants from non-perturbative topological recursion of A-polynomials, preprint, arXiv:1205.2261.

  8. 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 

  9. G. Borot and A. Guionnet, Asymptotic expansion of \({\beta}\) matrix models in the multi-cut regime, preprint, arXiv:1303.1045.

  10. G. Borot, A. Guionnet and K.K. Kozlowski, Large-N asymptotic expansion for mean field models with coulomb gas interaction, Int. Math. Res. Not. IMRN, 2015, 10451–10524.

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  13. L. Chekhov and B. Eynard, Matrix eigenvalue model: Feynman graph technique for all genera, J. High Energy Phys., 2006, no. 12, 026.

  14. B. Collins, Moments and cumulants of polynomial random variables on unitary groups, the Itzykson–Zuber integral, and free probability, Int. Math. Res. Not., 2003, 953–982.

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

    Article  MathSciNet  MATH  Google Scholar 

  16. Dabrowski Y.: A note about proving non-\({\Gamma}\) under a finite non-microstates free Fisher information assumption, J. Funct. Anal., 258, 3662–3674 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  17. Y. Dabrowski, Analytic functions relative to a covariance map \({\eta}\): I. Generalized Haagerup products and analytic relations, preprint, arXiv:1503.05515.

  18. Y. Dabrowski, A. Guionnet and D. Shlyakhtenko, Free transport for convex potentials, preprint, 2016.

  19. Dabrowski Y., Ioana A.: Unbounded derivations, free dilations, and indecomposability results for II\({_1}\) factors. Trans. Amer. Math. Soc., 368, 4525–4560 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  20. P. Deift and D. Gioev, Random Matrix Theory: Invariant Ensembles and Universality, Courant Lect. Notes Math., 18, Courant Inst. Math. Sci., New York, NY, 2009.

  21. Eynard B., Orantin N.: Invariants of algebraic curves and topological expansion. Commun. Number Theory Phys., 1, 347–452 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  22. A. Guionnet, Large Random Matrices: Lectures on Macroscopic Asymptotics. Lectures from the 36th Probability Summer School held in Saint-Flour, 2006, Lecture Notes in Math., 1957, Springer-Verlag, 2009.

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

  24. 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 

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

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  27. A. Guionnet and J. Novak, Asymptotics of unitary multimatrix models: The Schwinger–Dyson lattice and topological recursion, preprint, arXiv:1401.2703.

  28. Guionnet A., Shlyakhtenko D.: Free diffusions and matrix models with strictly convex interaction. Geom. Funct. Anal., 18, 1875–1916 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  30. Haagerup U., Thorbjørnsen S.: A new application of random matrices: \({{\rm Ext}(C^*_{\rm red}(F_2))}\) is not a group. Ann. of Math., (2) 162, 711– (2005)

    Article  MathSciNet  MATH  Google Scholar 

  31. Jones V.F.R., Shlyakhtenko D., Walker K.: An orthogonal approach to the subfactor of a planar algebra. Pacific J. Math., 246, 187–197 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  32. E. Maurel-Segala, High order expansion of matrix models and enumeration of maps, preprint, arXiv:math/0608192.

  33. N.A. Nekrasov, Non-perturbative Dyson–Schwinger equations and BPS/CFT correspondence, in preparation (2015).

  34. N.A. Nekrasov and V. Pestun, Seiberg–Witten geometry of four dimensional \({N = 2}\) quiver gauge theories, preprint, arXiv:1211.2240.

  35. N.A. Nekrasov, V. Pestun and S. Shatashvili, Quantum geometry and quiver gauge theories, preprint, arXiv:1312.6689 [hep-th].

  36. Nelson B.: Free monotone transport without a trace. Comm. Math. Phys., 334, 1245–1298 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  38. Ozawa N.: There is no separable universal \({{\rm II_{1}}}\)-factor. Proc. Amer. Math. Soc., 132, 487–490 (2004) (electronic)

    Article  MathSciNet  MATH  Google Scholar 

  39. Pastur L.: Limiting laws of linear eigenvalue statistics for Hermitian matrix models. J. Math. Phys., 47, 103303 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  40. 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  Google Scholar 

  41. M. Shcherbina, Change of variables as a method to study general \({\beta}\)-models: bulk universality, preprint, arXiv:1310.7835.

  42. 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 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  45. F.G. Tricomi, Integral Equations, Pure and Applied Mathematics. Vol. V, Interscience Publishers, Inc., New York, NY, 1957.

  46. Tutte W.T.: A census of planar maps. Canad. J. Math., 15, 249–271 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  47. D.-V. Voiculescu, The coalgebra of the free difference quotient and free probability, Internat. Math. Res. Notices, 2000, 79–106.

  48. Voiculescu D.-V.: A note on cyclic gradients. Indiana Univ. Math. J., 49, 837–841 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  49. Voiculescu D.-V.: Aspects of free analysis. Jpn. J. Math., 3, 163–183 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  50. A. Zvonkin, Matrix integrals and map enumeration: an accessible introduction, In: Combinatorics and Physics, Marseilles, 1995, Math. Comput. Modelling, 26, Elsevier, Amsterdam, 1997, pp. 281–304.

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Authors and Affiliations

  1. Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA, 02139-4307, USA

    Alice Guionnet

  2. Centre national de la recherche scientifique, 3, rue Michel-Ange, 75794, Paris cedex 16, France

    Alice Guionnet

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  1. Alice Guionnet
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Correspondence to Alice Guionnet.

Additional information

Communicated by: Yasuyuki Kawahigashi

This article is based on the 14th Takagi Lectures that the author delivered at University of Tokyo on November 15 and 16, 2014.

This work was partially supported by the NSF and Simons foundation.

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Guionnet, A. Free analysis and random matrices. Jpn. J. Math. 11, 33–68 (2016). https://doi.org/10.1007/s11537-016-1489-1

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  • DOI: https://doi.org/10.1007/s11537-016-1489-1

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