## The Annals of Probability

### Extremal eigenvalue correlations in the GUE minor process and a law of fractional logarithm

#### Abstract

Let $\lambda^{(N)}$ be the largest eigenvalue of the $N\times N$ GUE matrix which is the $N$th element of the GUE minor process, rescaled to converge to the standard Tracy–Widom distribution. We consider the sequence $\{\lambda^{(N)}\}_{N\geq1}$ and prove a law of fractional logarithm for the $\limsup$:

$\limsup_{N\to\infty}\frac{\lambda^{({{N}})}}{(\log N)^{2/3}}=(\frac{1}{4})^{2/3}\qquad \mbox{almost surely}.$ For the $\liminf$, we prove the weaker result that there are constants $c_{1},c_{2}>0$ so that

$-c_{1}\leq\liminf_{N\to\infty}\frac{\lambda^{({{N}})}}{(\log N)^{1/3}}\leq-c_{2}\qquad \mbox{almost surely}.$ We conjecture that in fact, $c_{1}=c_{2}=4^{1/3}$.

#### Article information

Source
Ann. Probab., Volume 45, Number 6A (2017), 4112-4166.

Dates
Revised: October 2016
First available in Project Euclid: 27 November 2017

https://projecteuclid.org/euclid.aop/1511773674

Digital Object Identifier
doi:10.1214/16-AOP1161

Mathematical Reviews number (MathSciNet)
MR3729625

Zentralblatt MATH identifier
06838117

#### Citation

Paquette, Elliot; Zeitouni, Ofer. Extremal eigenvalue correlations in the GUE minor process and a law of fractional logarithm. Ann. Probab. 45 (2017), no. 6A, 4112--4166. doi:10.1214/16-AOP1161. https://projecteuclid.org/euclid.aop/1511773674

#### References

• [1] Anderson, G. W., Guionnet, A. and Zeitouni, O. (2010). An Introduction to Random Matrices. Cambridge Studies in Advanced Mathematics 118. Cambridge Univ. Press, Cambridge.
• [2] Borodin, A. (2014). CLT for spectra of submatrices of Wigner random matrices. Mosc. Math. J. 14 29–38, 170.
• [3] Borodin, A., Ferrari, P. L. and Sasamoto, T. (2008). Transition between $\mathrm{Airy}_{1}$ and $\mathrm{Airy}_{2}$ processes and TASEP fluctuations. Comm. Pure Appl. Math. 61 1603–1629.
• [4] Borodin, A. and Gorin, V. (2015). General $\beta$-Jacobi corners process and the Gaussian free field. Comm. Pure Appl. Math. 68 1774–1844.
• [5] Bourgade, P., Erdős, L., Yau, H.-T. and Yin, J. (2016). Fixed energy universality for generalized Wigner matrices. Comm. Pure Appl. Math. 69 1815–1881.
• [6] Deift, P., Its, A. and Krasovsky, I. (2008). Asymptotics of the Airy-kernel determinant. Comm. Math. Phys. 278 643–678.
• [7] Erdős, L., Ramírez, J., Schlein, B., Tao, T., Vu, V. and Yau, H.-T. (2010). Bulk universality for Wigner Hermitian matrices with subexponential decay. Math. Res. Lett. 17 667–674.
• [8] Erdős, L. and Yau, H.-T. (2012). A comment on the Wigner–Dyson–Mehta bulk universality conjecture for Wigner matrices. Electron. J. Probab. 17 no. 28, 5.
• [9] Ferrari, P. L. (2008). The universal $\mathrm{Airy}_{1}$ and $\mathrm{Airy}_{2}$ processes in the totally asymmetric simple exclusion process. In Integrable Systems and Random Matrices. Contemp. Math. 458 321–332. Amer. Math. Soc., Providence, RI.
• [10] Fleming, B. J., Forrester, P. J. and Nordenstam, E. (2012). A finitization of the bead process. Probab. Theory Related Fields 152 321–356.
• [11] Forrester, P. J. (2010). Log-Gases and Random Matrices. London Mathematical Society Monographs Series 34. Princeton Univ. Press, Princeton, NJ.
• [12] Forrester, P. J. and Nagao, T. (2008). Determinantal correlations for classical projection processes. Preprint. Available at arXiv:0801.0100.
• [13] Gohberg, I., Goldberg, S. and Krupnik, N. (2000). Traces and Determinants of Linear Operators. Operator Theory: Advances and Applications 116. Birkhäuser Verlag, Basel.
• [14] Gorin, V. and Panova, G. (2015). Asymptotics of symmetric polynomials with applications to statistical mechanics and representation theory. Ann. Probab. 43 3052–3132.
• [15] Hartman, P. and Wintner, A. (1941). On the law of the iterated logarithm. Amer. J. Math. 63 169–176.
• [16] Hough, J. B., Krishnapur, M., Peres, Y. and Virág, B. (2006). Determinantal processes and independence. Probab. Surv. 3 206–229.
• [17] Johansson, K. (2003). Discrete polynuclear growth and determinantal processes. Comm. Math. Phys. 242 277–329.
• [18] Johansson, K. and Nordenstam, E. (2006). Eigenvalues of GUE minors. Electron. J. Probab. 11 1342–1371.
• [19] Johansson, K. and Nordenstam, E. (2007). Erratum to: “Eigenvalues of GUE minors” [Electron. J. Probab. 11 (2006), no. 50, 1342–1371; MR2268547]. Electron. J. Probab. 12 1048–1051.
• [20] Kalai, G. (2013). Laws of iterated logarithm for random matrices and random permutation. Available at http://mathoverflow.net/questions/142371/laws-of-iterated-logarithm-for-random-matrices-and-random-permutation.
• [21] Ledoux, M. and Rider, B. (2010). Small deviations for beta ensembles. Electron. J. Probab. 15 1319–1343.
• [22] NIST Digital Library of Mathematical Functions. Available at http://dlmf.nist.gov/, Release 1.0.6 of 2013-05-06. Online companion to [24].
• [23] Okounkov, A. and Reshetikhin, N. (2006). The birth of a random matrix. Mosc. Math. J. 6 553–566, 588.
• [24] Olver, F. W. J., Lozier, D. W., Boisvert, R. F. and Clark, C. W., eds. (2010). NIST Handbook of Mathematical Functions. Cambridge Univ. Press, New York, NY. Print companion to [22].
• [25] Prähofer, M. and Spohn, H. (2002). Scale invariance of the PNG droplet and the Airy process. J. Stat. Phys. 108 1071–1106.
• [26] Simon, B. (1977). Notes on infinite determinants of Hilbert space operators. Adv. Math. 24 244–273.
• [27] Skovgaard, H. (1959). Asymptotic forms of Hermite polynomials. Technical report 18, Department of Mathematics, California Institute of Technology.
• [28] Tao, T. and Vu, V. (2011). The Wigner–Dyson–Mehta bulk universality conjecture for Wigner matrices. Electron. J. Probab. 16 2104–2121.
• [29] Tracy, C. A. and Widom, H. (1994). Level-spacing distributions and the Airy kernel. Comm. Math. Phys. 159 151–174.
• [30] Wei, S. (2008). A globally uniform asymptotic expansion of the Hermite polynomials. Acta Math. Sci. Ser. B Engl. Ed. 28 834–842.