In this paper, we study the random matrix model of Gaussian Unitary Ensemble (GUE) with fixed-rank (aka spiked) external source. We will focus on the critical regime of the Baik–Ben Arous–Péché (BBP) phase transition and establish the distribution of the eigenvectors associated with the leading eigenvalues. The distribution is given in terms of a determinantal point process with extended Airy kernel. Our result can be regarded as an eigenvector counterpart of the BBP eigenvalue phase transition [6]. The derivation of the distribution makes use of the recently re-discovered eigenvector–eigenvalue identity, together with the determinantal point process representation of the GUE minor process with external source.

在本文中，我们研究了具有固定秩（又名尖峰）外部源的高斯酉系综 (GUE) 的随机矩阵模型。我们将关注 Baik-Ben Arous-Péché (BBP) 相变的临界状态，并建立与主要特征值相关的特征向量的分布。该分布是根据具有扩展艾里核的行列式点过程给出的。我们的结果可以看作是 BBP 特征值相变的特征向量对应物 [6]。分布的推导利用了最近重新发现的特征向量-特征值恒等式，以及具有外部源的 GUE 次要过程的行列式点过程表示。