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The law of large numbers for the maximum of almost Gaussian log-correlated fields coming from random matrices

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).

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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.

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Correspondence to Gaultier Lambert.

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.

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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

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  • DOI: https://doi.org/10.1007/s00440-018-0832-2

Mathematics Subject Classification

  • 60B20
  • 60G70
  • 35Q15