Statistics of Extremes in Eigenvalue-Counting Staircases.

Abstract

We consider the number N_{θ_{A}}(θ) of eigenvalues e^{iθ_{j}} of a random unitary matrix, drawn from CUE_{β}(N), in the interval θ_{j}∈[θ_{A},θ]. The deviations from its mean, N_{θ_{A}}(θ)-E[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 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≠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.