论文标题
特征值计数楼梯中极端的统计数据
Statistics of extremes in eigenvalue-counting staircases
论文作者
论文摘要
我们考虑eigenValues的数字$ {\ cal n} _ {θ_a}（θ）$ $ e^{i^{i thantry单位矩阵的$ e^{iθ_j} $，从cue $_β（n）$中绘制为Interval $θ_j\ in [θ_A，θ_A，θ_A，θ_A，θ_A] $。偏离其平均值，$ {\ cal n} _ {θ_a}（θ） - \ Mathbb {e}（{\ cal n} _ {θ_a}（θ）（θ））$，形成一个随机过程，作为$θ$的函数。我们通过利用映射到对数相关的随机景观的统计力学来研究此过程的最大值。通过使用扩展的Fisher-hartwig猜想作为toeplitz决定因素，并补充了对数相关场的冻结二元性猜想，我们获得了任何$β> 0 $的最大分布的累积分布。它在$β= 2 $的$β= 2 $中免费的标准计数统计数据的合并特征，并以$β\ ne 2 $为$β\ ne 2 $的相互作用与赫斯特索引$ h = 0 $的分数布朗运动的间隔和极端统计。 $β= 2 $结果有望适用于Riemann Zeta功能的零统计数据
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
