Eigenvector distribution in the critical regime of BBP transition. (English) Zbl 1482.15036
Summary: 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 [J. Baik et al., Ann. Probab. 33, No. 5, 1643–1697 (2005; Zbl 1086.15022)]. 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.
MSC:
| 15B52 | Random matrices (algebraic aspects) |
| 60B20 | Random matrices (probabilistic aspects) |
| 15A18 | Eigenvalues, singular values, and eigenvectors |
| 62E20 | Asymptotic distribution theory in statistics |
Keywords:
eigenvector distribution; spiked random matrices; BBP transition; eigenvector-eigenvalue identity; GUE minor process; extended Airy kernalCitations:
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