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On the maximum of the C\(\beta\)E field. (English) Zbl 1457.60008

In this paper, the extremal values of (the logarithm of) the characteristic polynomial of a random unitary matrix whose spectrum is distributed according to the circular beta ensemble (\(C\beta E\)) are investigated. More precisely, assuming that \(X_n\) is this characteristic polynomial and \(\mathbb{U}\) is the unit circle, it is proved that \[\sup\limits_{z\in\mathbb{U}}\mathrm{Re} \log{X_n(z)}=\sqrt{\frac{2}{\beta}}\left(\log n-{\frac{3}{4}} \log{\log{n}}+\mathcal{O}(1)\right)\] as well as an analogous statement for the imaginary part. The notation \(\mathcal{O}(1)\) means that the corresponding family of random variables, indexed by \(n\), is tight. This answers a conjecture of [Y. Fyodorov et al., “Freezing transition, characteristic polynomials of random matrices, and the Riemann zeta function”, Phys. Rev. Lett., 108, No. 17, Article ID 170601, 5 p. (2012; doi:10.1103/PhysRevLett.108.170601); Y. Fyodorov and J. Keating, Philos. Trans. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 372, No. 2007, Article ID 20120503, 32 p. (2014; Zbl 1330.82028)]), originally formulated for the \(\beta=2\) case, which corresponds to the circular unitary ensemble (\(CUE\)) field.

MSC:

60B20 Random matrices (probabilistic aspects)
60G70 Extreme value theory; extremal stochastic processes
15B52 Random matrices (algebraic aspects)

Citations:

Zbl 1330.82028
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References:

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