Let $f_N(z)=\log |\det (zI-U_N)|$ where $U_N$ is a random unitary matrix and consider the random variable $Z_N=\max_{z: |z|=1} f_N(z)$. $f_N(\cdot)$, being a linear statistic of the eigenvalues of $U_N$, converges in distribution to a logarithmically correlated Gaussian field on the unit circle. Fyodorov, Gaith and Keating conjectured that $Z_N-\log N+(3/4) \log\log N$ converges in distribution and identified the conjectured limit. We verify their conjecture up to the second order term, that is we show that $(Z_N-\log N)/\log\log N\to 3/4$. Joint work with Elliot Paquette.
