Abstract
In this article, we investigate 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 (CE). More precisely, assuming that is this characteristic polynomial and is the unit circle, we prove that
as well as an analogous statement for the imaginary part. The notation means that the corresponding family of random variables, indexed by , is tight. This answers a conjecture of Fyodorov, Hiary, and Keating, originally formulated for the case, which corresponds to the circular unitary ensemble (CUE) field.
Citation
Reda Chhaibi. Thomas Madaule. Joseph Najnudel. "On the maximum of the CE field." Duke Math. J. 167 (12) 2243 - 2345, 1 September 2018. https://doi.org/10.1215/00127094-2018-0016
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