Delocalization of eigenvectors of random matrices with independent entries. (English) Zbl 1352.60007

Rudelson, Mark; Vershynin, Roman

Duke Math. J. 164, No. 13, 2507-2538 (2015).

Summary: We prove that an \(n\times n\) random matrix \(G\) with independent entries is completely delocalized. Suppose that the entries of \(G\) have zero means, variances uniformly bounded below, and a uniform tail decay of exponential type. Then with high probability all unit eigenvectors of \(G\) have all coordinates of magnitude \(O(n-1/2)\), modulo logarithmic corrections. This comes as a consequence of a new, geometric approach to delocalization for random matrices.

MSC:

60B20	Random matrices (probabilistic aspects)
15B52	Random matrices (algebraic aspects)

Keywords:

delocalization of eigenvectors; random matrices; rotation-invariant ensembles

References:

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