Statistics of extremes in eigenvalue-counting staircases

Abstract

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

Main text: 7 pages, Supp. Mat. 7 pages. 4 figures. Misprint corrected, references added, main text shortened, Supp. Mat. upgraded

References in corpus (12)

• Full counting statistics in a propagating quantum front and random matrix spectra
• Freezing and extreme value statistics in a Random Energy Model with logarithmically correlated potential
• Non-interacting fermions at finite temperature in a $d$-dimensional trap: universal correlations
• Phase transitions and edge scaling of number variance in Gaussian random matrices
• Random matrices and entanglement entropy of trapped Fermi gases
• Characterizing correlations with full counting statistics: classical Ising and quantum XY spin chains
• Full counting statistics in the Haldane-Shastry chain
• Freezing and decorated Poisson point processes
• Fredholm determinants, full counting statistics and Loschmidt echo for domain wall profiles in one-dimensional free fermionic chains
• Matrix models for circular ensembles
• On the partition function of the Riemann zeta function, and the Fyodorov--Hiary--Keating conjecture
• One step replica symmetry breaking and extreme order statistics of logarithmic REMs

Cited by in corpus (5)

• Counting statistics for non-interacting fermions in a $d$-dimensional potential
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• Maxima of log-correlated fields: some recent developments
• Index of a matrix, complex logarithms, and multidimensional Fresnel integrals