Work

Statistics of Extremes in Eigenvalue-Counting Staircases

Year: 2020

Type: article

Abstract: We consider the number ${\mathcal{N}}_{{\ensuremath{\theta}}_{A}}(\ensuremath{\theta})$ of eigenvalues ${e}^{i{\ensuremath{\theta}}_{j}}$ of a random unitary matrix, drawn from ${\mathrm{CUE}}_{\ensuremath{\beta}}(N)$, in the interval ${\ensuremath{\theta}}_{j}\ensuremath{\in}[{\ensuremath{\theta}}_{A},\ensuremath{\theta}]$. The deviations from its mean, ${\mathcal{N}}_{{\ensuremath{\theta}}_{A}}(\ensuremath{\theta})\ensuremath{-}\mathbb{E}[{\mathcal{N}}_{{\ensuremath{\theta}}_{A}}(\ensuremath{\theta})]$, form a random process as function of $\ensuremath{\theta}$. 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 $\ensuremath{\beta}>0$. It exhibits combined features of standard counting statistics of fermions (free for $\ensuremath{\beta}=2$ and with Sutherland-type interaction for $\ensuremath{\beta}\ensuremath{\ne}2$) in an interval and extremal statistics of the fractional Brownian motion with Hurst index $H=0$. The $\ensuremath{\beta}=2$ results are expected to apply to the statistics of zeroes of the Riemann Zeta function.

Source: Physical Review Letters

DOI: 10.1103/physrevlett.124.210602

Authors: Yan V. Fyodorov, Pierre Le Doussal