The Maximum of the CUE Field

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.