For Gaussian random matrices, the squared overlaps between singular vectors of a submatrix and of the full matrix have explicit limiting Cauchy-like formulas in the Marchenko-Pastur regime.
Random matrix minor processes related to percolation theory.Random Matrices: Theory and Applications, 2(04):1350008, 2013
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Eigenvector Overlaps of Random Covariance Matrices and their Submatrices
For Gaussian random matrices, the squared overlaps between singular vectors of a submatrix and of the full matrix have explicit limiting Cauchy-like formulas in the Marchenko-Pastur regime.