论文标题
sublinear等级的球形积分
Spherical Integrals of Sublinear Rank
论文作者
论文摘要
当$ k = o（n）$时，我们考虑$ k $二维球形积分的渐近学。我们证明$ o（n）$ - 尺寸球形积分大约是$ 1 $维的球形积分。我们的公式扩展了Guionnet和Maïda在[29]中证明的$ k $维球面积分的结果，[34]中的Husson和Guionnet仅适用于$ k $有效，独立于$ n $。这些近似值将用于证明与尖锐的副高斯木质矩阵的连接$ 2K（n）$ Extreme eigenvalues和GOE/GUE矩阵的添加变形的巨大偏差原理。此外，当旋转或信号的尺寸具有均方根生长时，我们的结果将用于计算球形SK矢量旋转玻璃的自由能以及矩阵估计问题的共同信息。
We consider the asymptotics of $k$-dimensional spherical integrals when $k = o(N)$. We prove that the $o(N)$-dimensional spherical integrals are approximately the products of $1$-dimensional spherical integrals. Our formulas extend the results for $k$-dimensional spherical integrals proved by Guionnet and Maïda in [29] and Husson and Guionnet in [34] which are only valid for $k$ finite and independent of $N$. These approximations will be used to prove a large deviation principle for the joint $2k(N)$ extreme eigenvalues for sharp sub-Gaussian Wigner matrices and for additive deformations of GOE/GUE matrices. Furthermore, our results will be used to compute the free energies of spherical SK vector spin glasses and the mutual information for matrix estimation problems when the dimensions of the spins or signals have sublinear growth.
