The Maximum of the CUE Field

Abstract Let $U_N$ denote a Haar Unitary matrix of dimension $N,$ and consider the field $\mathbf{U}_{{N}}(z) = \log |\det(1-zU_N)|$ for $z\in \mathbb{C}$. Then, $\frac{\max_{|z|=1} \mathbf{U}_{{N}}(z) - \log N + \frac{3}{4} \log\log N}{ \log\log N} \to 0 $ in probability. This provides a verification up to second order of a conjecture of Fyodorov, Hiary, and Keating, improving on the recent first order verification of Arguin, Belius and Bourgade.