Abstract
We compute the leading asymptotics as \(N\rightarrow \infty \) of the maximum of the field \(Q_N(q)= \log |q- A_N|, q\in \mathbb {C}\), for any unitarily invariant Hermitian random matrix \(A_N\) associated to a non-critical real-analytic potential. Hence, we verify the leading order in a conjecture of Fyodorov and Simm (Nonlinearity 29:2837, 2016. arXiv:1503.07110 [math-ph]) formulated for the GUE. The method relies on a classical upper-bound and a more sophisticated lower-bound based on a variant of the second-moment method which exploits the hyperbolic branching structure of the field \(Q_N(q), q\in \mathbb {H}\). Specifically, we compare \(Q_N\) to an idealized Gaussian field by means of exponential moments. In principle, this method could also be applied to random fields coming from other point processes provided that one can compute certain mixed exponential moments. For unitarily invariant ensembles, we show that these assumptions follow from the Fyodorov–Strahov formula (Fyodorov and Strahov in J Phys A 36(12):3203–3213, 2003. https://doi.org/10.1088/0305-4470/36/12/320) and asymptotics of orthogonal polynomials derived in Deift et al. (Commun Pure Appl Math 52(11):1335–1425, 1999. https://doi.org/10.1002/(SICI)1097-0312(199911)52:11<1335::AID-CPA1>3.0.CO;2-1).
This is a preview of subscription content, access via your institution.
References
Arguin, L.-P., Belius, D., Bourgade, P.: Maximum of the characteristic polynomial of random unitary matrices. In: Communications in Mathematical Physics (2016). arXiv:1511.07399 [math.PR]
Bai, Z.D., Silverstein, J.W.: CLT for linear spectral statistics of large-dimensional sample covariance matrices. Ann. Prob. 32(1A), 553–605 (2004). https://doi.org/10.1214/aop/1078415845
Borot, G., Guionnet, A.: Asymptotic expansion of beta matrix models in the multi-cut regime. In: ArXiv e-prints (2013). arXiv:1303.1045 [math-ph]
Chhaibi, R., Najnudel, J., Madaule, T.: On the maximum of the C\(\beta \)E field. In: ArXiv e-prints (2016). arXiv:1607.00243 [math.PR]
Daviaud, O.: Extremes of the discrete two-dimensional Gaussian free field. Ann. Prob. 34(3), 962–986 (2006). https://doi.org/10.1214/009117906000000061
de Monvel, A.B., Pastur, L., Shcherbina, M.: On the statistical mechanics approach in the random matrix theory: integrated density of states. J. Stat. Phys. 79(3–4), 585–611 (1995). https://doi.org/10.1007/BF02184872
Deift, P., Kriecherbauer, T., McLaughlin, K.T.-R., Venakides, S., Zhou, X.: Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory. Commun. Pure Appl. Math. 52(11), 1335–1425 (1999). https://doi.org/10.1002/(SICI)1097-0312(199911)52:11<1335::AID-CPA1>3.0.CO;2-1
Ding, J., Roy, R., Zeitouni, O.: Convergence of the centered maximum of log-correlated Gaussian fields. In: ArXiv e-prints (2015). arXiv:1503.04588 [math.PR]
Fyodorov, Y.V., Simm, N.J.: On the distribution of the maximum value of the characteristic polynomial of GUE random matrices. In: Nonlinearity 29, 2837 (2016). arXiv:1503.07110 [math-ph]
Fyodorov, Y.V., Strahov, E.: An exact formula for general spectral correlation function of random Hermitian matrices. J. Phys. A 36(12), 3203–3213 (2003). https://doi.org/10.1088/0305-4470/36/12/320. Random matrix theory
Fyodorov, Y.V., Hiary, G.A., Keating, J.P.: Freezing transition, characteristic polynomials of random matrices, and the Riemann zeta function. Phys. Rev. Lett. 108, 170601 (2012). https://doi.org/10.1103/PhysRevLett.108.170601
Hu, X., Miller, J., Peres, Y.: Thick points of the Gaussian free field. Ann. Prob. 38(2), 896–926 (2010). https://doi.org/10.1214/09-AOP498
Johansson, K.: On fluctuations of eigenvalues of random Hermitian matrices. Duke Math. J. 91(1), 151–204 (1998). https://doi.org/10.1215/S0012-7094-98-09108-6
Kuijlaars, A.B.J.: Riemann–Hilbert analysis for orthogonal polynomials. In: Koelink, E., Van Assche, W. (eds.) Orthogonal Polynomials and Special Functions: Leuven 2002, pp. 167–210. Springer, Berlin, Heidelberg (2003). https://doi.org/10.1007/3-540-44945-0_5
Paquette, E., Zeitouni, O.: The maximum of the CUE field. In: ArXiv e-prints (2016). arXiv:1602.08875 [math.PR]
Saksman, E., Webb, C.: The Riemann zeta function and Gaussian multiplicative chaos: statistics on the critical line. In: ArXiv e-prints (2016). arXiv:1609.00027 [math.PR]
Shcherbina, M.: Fluctuations of linear eigenvalue statistics of \(\beta \) matrix models in the multi-cut regime. J. Stat. Phys. 151(6), 1004–1034 (2013). https://doi.org/10.1007/s10955-013-0740-x
Tao, T., Vu, V.: Random matrices: universality of local eigenvalue statistics. Acta Math. 206(1), 127 (2011). https://doi.org/10.1007/s11511-011-0061-3
Webb, C.: The characteristic polynomial of a random unitary matrix and Gaussian multiplicative chaos—the \(L^{2}\)-phase. Electron. J. Prob. 20(104), 21 (2015). https://doi.org/10.1214/EJP.v20-4296
Acknowledgements
We would like to thank Ofer Zeitouni and Kurt Johansson for helpful conversation. We would especially like to thank Ofer for pointing out the Fyodorov–Strahov formula which started this project.
Author information
Authors and Affiliations
Corresponding author
Additional information
G.L. gratefully acknowledges the support of the grant KAW 2010.0063 from the Knut and Alice Wallenberg Foundation while at KTH, Royal Institute of Technology. E.P. gratefully acknowledges the support of NSF Postdoctoral Fellowship DMS-1304057. This work began while G.L. visited Weizmann, supported in part by a grant from the Israel Science Foundation. E.P. would also like to thank Kurt Johansson for his invitation to visit KTH, during which time part of this work was completed.
Rights and permissions
About this article
Cite this article
Lambert, G., Paquette, E. The law of large numbers for the maximum of almost Gaussian log-correlated fields coming from random matrices. Probab. Theory Relat. Fields 173, 157–209 (2019). https://doi.org/10.1007/s00440-018-0832-2
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00440-018-0832-2
Mathematics Subject Classification
- 60B20
- 60G70
- 35Q15