2016/07/31 by Reda Chhaibi, Thomas Madaule, Joseph Najnudel
Mathematics · Physics and Astronomy · #Advanced Combinatorial Mathematics #Analytic Number Theory Research #Characteristic polynomial #Combinatorics #Conjecture #Discrete mathematics #Field (mathematics) #Finite field #Logarithm #Mathematical analysis #Mathematics #Physics #Polynomial #Pure mathematics #Quantum mechanics #Random Matrices and Applications #Spectrum (functional analysis) #Unit (ring theory) #Unit circle #Unitary matrix #Unitary state #math-ph #math.CA #math.CV #math.MP #math.PR
paper · pdf · doi:10.1215/00127094-2018-0016
published as Duke Math. J. 167, no. 12 (2018), 2243-2345 · 74 pages ; v1: Preliminary version; v2: Submitted version
openalex created_date 2016/08/23 · arxiv created 2017/01/04 · openalex publication_date 2018/08/10 · arxiv updated 2018/11/14 · openalex updated_date 2026/08/05
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 (CβE). More precisely, assuming that Xn is this characteristic polynomial and U is the unit circle, we prove that sup z∈UℜlogXn(z)=2β(logn−34loglogn+O(1)), as well as an analogous statement for the imaginary part. The notation O(1) means that the corresponding family of random variables, indexed by n, is tight. This answers a conjecture of Fyodorov, Hiary, and Keating, originally formulated for the β=2 case, which corresponds to the circular unitary ensemble (CUE) field.