Small Gaps of GOE


In this article, we study the smallest gaps between eigenvalues of the Gaussian orthogonal ensemble (GOE). The main result is that the smallest gaps, after being normalized by n, will converge to a Poisson distribution, and the limiting density of the kth normalized smallest gap is \(2{}x^{2k-1}e^{-x^{2}}/(k-1)!\). The proof is based on the method developed in Feng and Wei (Small gaps of circular \(\beta \)-ensemble. arXiv:1806.01555). We need to prove the convergence of the factorial moments of the smallest gaps, which makes use of the Pfaffian structure of GOE and some comparison results between the one-component log-gas and the two-component log-gas.

This is a preview of subscription content, access via your institution.

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.


  1. AGZ10

    G.W. Anderson, A. Guionnet, and O. Zeitouni. An Introduction to Random Matrices. Cambridge Studies in Advanced Mathematics, vol. 118. Cambridge University Press, Cambridge (2010).

  2. BT77

    M.V. Berry and M. Tabor. Level clustering in the regular spectrum. Proc. R. Soc. London A 356 (1977), 375–394.

    Article  Google Scholar 

  3. BB13

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

    MathSciNet  Article  Google Scholar 

  4. BBRR17

    V. Blomer, J. Bourgain, M. Radziwill, and Z. Rudnick. Small gaps in the spectrum of the rectangular billiard. Ann. Sci. Éc. Norm. Supér (4) (5)50 (2017), 1283–1300.

    MathSciNet  Article  Google Scholar 

  5. BGS86

    O. Bohigas, M.-J. Giannoni, and C. Schmit. Spectral fluctuations of classically chaotic quantum systems. In: Quantum Chaos and Statistical Nuclear Physics, edited by Thomas H. Seligman and Hidetoshi Nishioka, Lecture Notes in Physics, vol. 263. Springer, Berlin (1986), pp. 18–40.

  6. Bou

    P. Bourgade. Extreme gaps between eigenvalues of Wigner matrices, arXiv:1812.10376.

  7. BEY14

    P. Bourgade, L. Erdős, and H.-T. Yau. Universality of general \(\beta \)-ensembles. Duke Math. J. (6)163 (2014), 1127–1190.

    MathSciNet  Article  Google Scholar 


    J. S. Cotler, G. Gur-Ari, M. Hanada, J. Polchinski, P. Saad, S.H. Shenker, D. Stanford, A. Streicher, and M. Tezuka. Black holes and random matrices. J. High Energy. Phys. 2017 (2017), 118.

    MathSciNet  Article  Google Scholar 

  9. DG09

    P. Deift and D. Gioev. Random Matrix Theory: Invariant Ensembles and Universality. Courant Lecture Notes Volume: 18 (2009) 217 pp, AMS.

  10. Dia03

    P. Diaconis. Patterns in eigenvalues: The 70th Josiah Willard Gibbs lecture. Bull. Am. Math. Soc. (N.S.) 40 (2003), 155–178.

    MathSciNet  Article  Google Scholar 

  11. EY15

    L. Erdős and H.-T. Yau. Gap universality of generalized Wigner and \(\beta \)-ensembles. J. Eur. Math. Soc. (JEMS) (8)17 (2015), 1927–2036.

    MathSciNet  Article  Google Scholar 

  12. FTWa

    R. Feng, G. Tian, and D. Wei. Spectrum of SYK model, arXiv:1801.10073, to appear in Peking Mathematical Journal.

    Article  Google Scholar 

  13. FTWb

    R. Feng, G. Tian, and D. Wei. Spectrum of SYK model II: Central limit theorem, arXiv: 1806.05714.

  14. FTWc

    R. Feng, G. Tian, and D. Wei. Spectrum of SYK model III: Large deviations and concentration of measures. arXiv:1806.04701.

  15. FWa

    R. Feng and D. Wei. Small gaps of circular \(\beta \)-ensemble. arXiv:1806.01555.

  16. FWb

    R. Feng and D. Wei. Large gaps of CUE and GUE. arXiv:1807.02149.

  17. FG16

    A. Figalli and A. Guionnet. Universality in several-matrix models via approximate transport maps. Acta Math. (1)217 (2016), 81–176.

    MathSciNet  Article  Google Scholar 

  18. For

    P.J. Forrester. Log-gases and random matrices, LMS-34, Princeton University Press.

  19. GV16

    A.M. Garcia-Garcia and J.J.M. Verbaarschot. Spectral and thermodynamic properties of the Sachdev-Ye-Kitaev model. Phys. Rev. D94 (2016), 126010.

    Article  Google Scholar 

  20. GV17

    A.M. Garcia-Garcia and J.J.M. Verbaarschot. Analytical spectral density of the Sachdev-Ye-Kitaev model at finite \(N\). Phys. Rev. D96 (2017), 066012.

    Google Scholar 

  21. KS99

    N.M. Katz and P. Sarnak. Random Matrices, Frobenius Eigenvalues and Monodromy, volume 45 of American Mathematical Society Colloquium Publications. American Mathematical Society, Providence, RI (1999).

  22. LLM

    B. Landon, P. Lopatto, and J. Marcinek. Comparison theorem for some extremal eigenvalue statistics. arXiv:1812.10022.

  23. Meh91

    M.L. Mehta. Random Matrices. Academic Press, New York, 2nd edition (1991).

    Google Scholar 

  24. NTV17

    H. Nguyen, T. Tao, and V. Vu. Random matrices: tail bounds for gaps between eigenvalues. Probab. Theory Relat. Fields (3-4)167 (2017), 777–816.

    MathSciNet  Article  Google Scholar 

  25. RSX13

    B. Rider, C.D. Sinclair, and Y. Xu. A solvable mixed charge ensemble on the line: global results. Probab. Theory Relat. Fields 155 (2013), 127–164.

    MathSciNet  Article  Google Scholar 

  26. STKZ

    M. Smaczynski, T. Tkocz, M. Kus, and K. Zyczkowski. Extremal spacings between eigenphases of random unitary matrices and their tensor products. Phys. Rev. E 88, 052902.

  27. Sos

    A. Soshnikov. Statistics of extreme spacing in determinantal random point processes. Mosc. Math. J. 5, 705–719, 744.

    MathSciNet  Article  Google Scholar 

  28. Tao13

    T. Tao. The asymptotic distribution of a single eigenvalue gap of a Wigner matrix. Probab. Theory Relat. Fields (1-2)157 (2013), 81–106.

    MathSciNet  Article  Google Scholar 

  29. TV11

    T. Tao and V. Vu. Random matrices: universality of local eigenvalue statistics. Acta Math. (1)206 (2011), 127–204.

    MathSciNet  Article  Google Scholar 

  30. Vin01

    J. Vinson, Closest spacing of eigenvalues. Ph.D. thesis, Princeton University (2001).

Download references


We would like to thank P. Bourgade, O. Zeitouni, G. Ben Arous and P. Forrester for many helpful discussions. We are indebted to the anonymous reviewers for providing many corrections and insightful comments, this paper would not have been possible without their supportive work.

Author information



Corresponding author

Correspondence to Renjie Feng.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Feng, R., Tian, G. & Wei, D. Small Gaps of GOE. Geom. Funct. Anal. 29, 1794–1827 (2019).

Download citation