Scaling limit for determinantal point processes on spheres
Scaling limit for determinantal point processes on spheres
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发表时间：
2022-03
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通讯作者：
M. Katori;T. Shirai
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作者：
M. Katori;T. Shirai
具有Haar概率测度的酉群称为循环酉包络。所有的特征值都落在复平面的单位圆上，它们可以看作是S上的一个行列式点过程。当矩阵的大小趋于∞时，标度点过程弱收敛于与所谓的正弦核相关的行列式点过程.我们将这一结果推广到高维球的情况，并表明标度极限过程是与第一类贝塞尔函数所表示的核相关联的决定点过程。
The unitary group with the Haar probability measure is called Circular Unitary Ensemble. All the eigenvalues lie on the unit circle in the complex plane and they can be regarded as a determinantal point process on S. It is also known that the scaled point processes converge weakly to the determinantal point process associated with the so-called sine kernel as the size of matrices tends to ∞. We extend this result to the case of high-dimensional spheres and show that the scaling limit processes are determinantal point processes associated with the kernels expressed by the Bessel functions of the first kind.