We study the volume of the intersection of two unit balls from one of the classical matrix ensembles GOE, GUE and GSE, as the dimension tends to infinity. This can be regarded as a matrix analogue of a result of Schechtman and Schmuckenschl\"ager for classical $\ell_p$-balls [Schechtman and Schmuckenschl\"ager, GAFA Lecture Notes, 1991]. The proof of our result is based on two ingredients, which are of independent interest. The first one is a weak law of large numbers for a point chosen uniformly at random in the unit ball of such a matrix ensemble. The second one is an explicit computation of the asymptotic volume of such matrix unit balls, which in turn is based on the theory of logarithmic potentials with external fields.

中文翻译：

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经典矩阵系综中单位球的交集

我们研究了来自经典矩阵集合 GOE、GUE 和 GSE 之一的两个单位球的交集的体积，因为维度趋于无穷大。这可以看作是 Schechtman 和 Schmuckenschl\"ager 对经典 $\ell_p$-balls [Schechtman and Schmuckenschl\"ager, GAFA Lecture Notes, 1991] 的结果的矩阵模拟。我们结果的证明基于两个具有独立意义的成分。第一个是弱大数定律，用于在这种矩阵系综的单位球中随机均匀选择的点。第二个是对这种矩阵单位球的渐近体积的显式计算，这又是基于具有外部场的对数势理论。