We obtain an asymptotic formula for \(n\times n\) Toeplitz determinants as \(n\rightarrow \infty \), for non-negative symbols with any fixed number of Fisher–Hartwig singularities, which is uniform with respect to the location of the singularities. As an application, we prove a conjecture by Fyodorov and Keating (Philos Trans R Soc A 372: 20120503, 2014) regarding moments of averages of the characteristic polynomial of the Circular Unitary Ensemble. In addition, we obtain an asymptotic formula regarding the momentum of impenetrable bosons in one dimension with periodic boundary conditions.

中文翻译：

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Fisher-Hartwig奇异性的Toeplitz行列式的一致渐近性

对于具有任意固定数量的Fisher–Hartwig奇点的非负符号，我们获得\（n \ timesn \） Toeplitz行列式为\（n \ rightarrow \ infty \）的渐近公式的奇点。作为应用，我们证明了Fyodorov和Keating（Philos Trans R Soc A 372：20120503，2014）关于圆Unit整体式特征多项式的平均矩的猜想。此外，我们获得了关于一维具有周期性边界条件的不可渗透的玻色子动量的渐近公式。