Local spectral statistics of the addition of random matrices

  • Ziliang CheEmail author
  • Benjamin Landon


We consider the local statistics of \(H = V^* X V + U^* Y U\) where V and U are independent Haar-distributed unitary matrices, and X and Y are deterministic real diagonal matrices. In the bulk, we prove that the gap statistics and correlation functions coincide with the GUE in the limit when the matrix size \(N \rightarrow \infty \) under mild assumptions on X and certain rigidity assumptions on Y (the latter being an assumption on the convergence of the eigenvalues of Y to the quantiles of its limiting spectral measure which we assume to have a density). Our method relies on running a carefully chosen diffusion on the unitary group and comparing the resulting eigenvalue process to Dyson Brownian motion. Our method also applies to the case when V and U are drawn from the orthogonal group. Our proof relies on the local law for H proved in Bao et al. (Commun Math Phys 349(3):947–990, 2017; J Funct Anal 271(3):672–719, 2016; Adv Math 319:251–291, 2017) as well as the DBM convergence results of Landon and Yau (Commun Math Phys 355(3):949–1000, 2017) and Landon et al. (Adv Math 346:1137–1332, 2019).

Mathematics Subject Classification

15B52 60B20 



The authors would like to thank H.-T. Yau and P. Sosoe for useful and enlightening discussions.


  1. 1.
    Adlam, B., Che, Z.: Spectral statistics of sparse random graphs with a general degree distribution. Preprint arXiv:1509.03368 (2015)
  2. 2.
    Ajanki, O., Erdös, L., Krüger, T.: Local eigenvalue statistics for random matrices with general short range correlations. Preprint arXiv:1604.08188
  3. 3.
    Ajanki, O., Erdős, L., Krüger, T.: Local semicircle law with imprimitive variance matrix. Electron. Commun. Probab. 19, 1–9 (2014)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Ajanki, O., Erdős, L., Krüger, T.: Quadratic vector equations on complex upper half-plane. Preprint arXiv:1506.05095 (2015)
  5. 5.
    Ajanki, O., Erdős, L., Krüger, T.: Universality for general Wigner-type matrices. Probab. Theory Relat. Fields 169, 667 (2017)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Anderson, G.W., Guionnet, A., Zeitouni, O.: An Introduction to Random Matrices, vol. 118. Cambridge University Press, Cambridge (2010)zbMATHGoogle Scholar
  7. 7.
    Bao, Z., Erdös, L., Schnelli, K.: Local law of addition of random matrices on optimal scale. Commun. Math. Phys. 349(3), 947–990 (2017) MathSciNetzbMATHGoogle Scholar
  8. 8.
    Bao, Z., Erdős, L., Schnelli, K.: Local stability of the free additive convolution. J. Funct. Anal. 271(3), 672–719 (2016)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Bao, Z., Erdös, L., Schnelli, K.: Convergence rate for spectral distribution of addition of random matrices. Adv. Math. 319, 251–291 (2017)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Bauerschmidt, R., Huang, J., Knowles, A., Yau, H.-T.: Bulk eigenvalues statistics for random regular graphs. Ann. Probab. 45(6A), 3626–3663 (2017)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Bauerschmidt, R., Knowles, A., Yau, H.-T.: Local semicircle law for random regular graphs. Commun. Pure Appl. Math. 70, 1898–1960 (2017)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Belinschi, S.T.: The Lebesgue decomposition of the free additive convolution of two probability distributions. Probab. Theory Relat. Fields 142(1–2), 125–150 (2006)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Belinschi, S.T.: A note on regularity for free convolutions. Ann. Inst. Henri Pointcarè Probab. Stat. 42(5), 635–648 (2006)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Belinschi, S.T., Bercovici, H.: A new approach to subordination results in free probability. J. Anal. Math. 101(1), 357–365 (2007)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Biane, P.: Representations of symmetric groups and free probability. Adv. Math. 138(1), 126–181 (1998)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Bobkov, S.G., GötzeA, F., Tikhomirov, N.: On concentration of empirical measures and convergence to the semi-circle law. J. Theor. Probab. 23, 792–823 (2010)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Bourgade, P., Erdős, L., Yau, H.-T., Yin, J.: Fixed energy universality for generalized Wigner matrices. Commun. Pure Appl. Math. 69, 1815–1881 (2015)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Bourgade, P., Erdős, L., Yau, H.-T., Yin, J.: Universality for a class of random band matrices. Adv. Theo. Math. Phys. 21(3), 739–800 (2017)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Bourgade, P., Yau, H.-T.: The eigenvector moment flow and local quantum unique ergodicity. Commun. Math. Phys. 350(1), 231–278 (2017)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Che, Z.: Universality of random matrices with correlated entries. Electr. J. Probab. 22(30), 1–38 (2017)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Collins, B.: Moments and cumulants of polynomial random variables on unitary groups, the Itzykson–Zuber integral, and free probability. Int. Math. Res. Not. 2003(17), 953–982 (2003)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Collins, B.: Product of random projections, Jacobi ensembles and universality problems arising from free probability. Probab. Theory Relat. Fields 133(3), 315–344 (2005)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Dyson, F.J.: A Brownian-motion model for the eigenvalues of a random matrix. J. Math. Phys. 3(6), 1191–1198 (1962)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Erdös, L., Knowles, A., Yau, H.-T., Yin, J.: Spectral statistics of Erdös–Rényi graphs II: eigenvalue spacing and the extreme eigenvalues. Commun. Math. Phys. 314(3), 587–640 (2012)zbMATHGoogle Scholar
  25. 25.
    Erdös, L., Knowles, A., Yau, H.-T., Yin, J.: Spectral statistics of Erdös–Rényi graphs I: local semicircle law. Ann. Probab. 41(3B), 2279–2375 (2013)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Erdős, L., Knowles, A., Yau, H.-T., Yin, J.: The local semicircle law for a general class of random matrices. Electron. J. Prob. 18(59), 1–58 (2013)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Erdös, L., Péché, S., Ramirez, J.A., Schlein, B.: Bulk universality for Wigner matrices. Commun. Pure Appl. Math. 63(7), 895–925 (2010)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Erdős, L., Schlein, B., Yau, H.-T., Yin, J., et al.: The local relaxation flow approach to universality of the local statistics for random matrices. 48(1):1–46 (2012)Google Scholar
  29. 29.
    Erdős, L., Schnelli, K.: Universality for random matrix flows with time-dependent density. Ann. Inst. H. Poincaré Probab. Statist. 53(4), 1606–1656 (2017)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Erdös, L., Yau, H.-T.: Gap universality of generalized Wigner and beta-ensembles. J. Eur. Math. 17(8), 1927–2036 (2015)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Erdös, L., Yau, H.-T., Schlein, B.: Unversality of random matrices and local relaxation flow. Invent. Math. 185(1), 75–119 (2011)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Erdös, L., Yau, H.-T., Yin, J.: Bulk universality for generalized Wigner matrices. Probab. Theory Relat. Fields 154(1–2), 341–407 (2012)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Erdős, L., Yau, H.-T., Yin, J.: Rigidity of eigenvalues of generalized Wigner matrices. Adv. Math. 229(3), 1435–1515 (2012)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Huang, J., Landon, B., Yau, H.-T.: Bulk universality of sparse random matrices. J. Math. Phys. 56(12), 123301 (2015)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Huang, J., Landon, B., Yau, H.-T.: Local law and mesoscopic fluctuations for Dyson Brownian motion with general \(\beta \) and potential. Preprint arXiv:1612.06306 (2016)
  36. 36.
    Karatzas, I., Shreve, S.E.: Browniam Motion and Stochastic Calculus. Springer, Berlin (1991)zbMATHGoogle Scholar
  37. 37.
    Kargin, V.: A concentration inequality and a local law for the sum of two random matrices. Probab. Theory Relat. Fields 154, 677–702 (2012)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Kargin, V.: On eigenvalues of the sum of two random projections. J. Stat. Phys. 149(2), 246–258 (2012)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Kargin, V.: Subordination for the sum of two random matrices. Ann. Probab. 43(4), 2119–2150 (2015)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Landon, B., Sosoe, P., Yau, H.-T.: Fixed energy universality for Dyson Brownian motion. Adv. Math. 346, 1137–1332 (2019)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Landon, B., Yau, H.-T.: Convergence of local statistics of Dyson Brownian motion. Commun. Math. Phys. 355(3), 949–1000 (2017)MathSciNetzbMATHGoogle Scholar
  42. 42.
    Lee, J.O., Schnelli, K.: Local deformed semicircle law and complete delocalization for Wigner matrices with random potential. J. Math. Phys. 54(10), 103504 (2013)MathSciNetzbMATHGoogle Scholar
  43. 43.
    Lee, J.O., Schnelli, K., Stetler, B., Yau, H.-T.: Bulk universality for deformed Wigner matrices. Ann. Probab. 44(3), 2349–2425 (2016)MathSciNetzbMATHGoogle Scholar
  44. 44.
    Maassen, H.: Addition of freely independent random variables. J. Funct. Anal. 106, 409–438 (1992)MathSciNetzbMATHGoogle Scholar
  45. 45.
    Oksendal, B.: Stochastic Differential Equations, 6th edn. Springer, Berlin (2007)zbMATHGoogle Scholar
  46. 46.
    Pastur, L., Vasilchuk, V.: On the law of addition of random matrices. Commun. Math. Phys. 214(2), 249–286 (2000)MathSciNetzbMATHGoogle Scholar
  47. 47.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics II: Fourier Analysis, Self-adjointness. Academic Press, London (1975)zbMATHGoogle Scholar
  48. 48.
    Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion, vol. 293. Springer, Berlin (2013)zbMATHGoogle Scholar
  49. 49.
    Speicher, R.: Free convolution and the random sum of matrices. Pub. Res. Inst. Math. 29(5), 731–744 (1993)MathSciNetzbMATHGoogle Scholar
  50. 50.
    Tao, T., Vu, V.: Random matrices: universality of local eigenvalue statistics. Commun. Math. Phys. 298(2), 549–572 (2010)MathSciNetzbMATHGoogle Scholar
  51. 51.
    Tao, T., Vu, V.: Random matrices: universality of local eigenvalue statistics up to the edge. Acta Math. 206(1), 127–204 (2011)MathSciNetzbMATHGoogle Scholar
  52. 52.
    Voiculescu, D.: Limit laws for random matrices and free products. Invent. Math. 104(1), 201–220 (1991)MathSciNetzbMATHGoogle Scholar
  53. 53.
    Voiculescu, D.: The analogues of entropy and of Fisher’s information measure in free probability theory, I. Commun. Math. Phys. 155(1), 71–92 (1993)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsHarvard UniversityCambridgeUSA

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