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Statistics of extremes in eigenvalue-counting staircases

We consider the number ${\cal N}_{θ_A}(θ)$ of eigenvalues $e^{i θ_j}$ of a random unitary matrix, drawn from CUE$_β(N)$, in the interval $θ_j \in [θ_A,θ]$. The deviations from its mean, ${\cal N}_{θ_A}(θ) - \mathbb{E}({\cal N}_{θ_A}(θ))$, form a random process as function of $θ$. We study the maximum of this process, by exploiting the mapping onto the statistical mechanics of log-correlated random landscapes. By using an extended Fisher-Hartwig conjecture for Toeplitz determinants, supplemented with the freezing duality conjecture for log-correlated fields, we obtain the cumulants of the distribution of that maximum for any $β>0$. It exhibits combined features of standard counting statistics of fermions (free for $β=2$ and with Sutherland-type interaction for $β\ne 2$) in an interval and extremal statistics of the fractional Brownian motion with Hurst index $H=0$. The $β=2$ results are expected to apply to the statistics of zeroes of the Riemann Zeta function