TY - JOUR

T1 - Statistics of Extremes in Eigenvalue-Counting Staircases

AU - Fyodorov, Yan V.

AU - Le Doussal, Pierre

PY - 2020/5/29

Y1 - 2020/5/29

N2 - We consider the number NθA(θ) of eigenvalues eiθj of a random unitary matrix, drawn from CUEβ(N), in the interval θj∈[θA,θ]. The deviations from its mean, NθA(θ)-E[NθA(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.

AB - We consider the number NθA(θ) of eigenvalues eiθj of a random unitary matrix, drawn from CUEβ(N), in the interval θj∈[θA,θ]. The deviations from its mean, NθA(θ)-E[NθA(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.

UR - http://www.scopus.com/inward/record.url?scp=85085987949&partnerID=8YFLogxK

U2 - 10.1103/PhysRevLett.124.210602

DO - 10.1103/PhysRevLett.124.210602

M3 - Article

AN - SCOPUS:85085987949

SN - 0031-9007

VL - 124

JO - Physical Review Letters

JF - Physical Review Letters

IS - 21

M1 - 210602

ER -