Eigenvector distribution of random matrices under critical finite-rank deformations

We investigate the eigenvector distribution at the soft edge for Gaussian random matrices with finite-rank deformations, in the critical regime of BBP transition. For finite-rank deformations of the GOE and GUE with critical spikes, we find that the squared overlap between a leading eigenvector and a spike, rescaled by $N^{1\mathrm{/}3}$, converges weakly to the negative reciprocal of the derivative of an Airy-Green function evaluated at the corresponding soft-edge root. For the rank-one critically spiked Gaussian $\beta$-ensemble, $\beta >0$, we obtain an analogous result involving an Airy-Green function. In both cases, the Airy-Green functions are generalizations of the one introduced by Bykhovskaya–Gorin–Sodin \cite{Bykhovskaya-Gorin-Sodin25}. The proofs are both based on an eigenvector–eigenvalue identity and a resolvent-differentiation mechanism.