TY - JOUR
TI - Optimal upper bound for the infinity norm of eigenvectors of random matrices
DO - https://doi.org/doi:10.7282/T3GB22N0
PY - 2013
AB - Let $M_n$ be a random Hermitian (or symmetric) matrix whose upper diagonal and diagonal entries are independent random variables with mean zero and variance one. It is well known that the empirical spectral distribution (ESD) converges in probability to the semicircle law supported on $[-2,2]$. In this thesis we study the local convergence of ESD to the semicircle law. One main result is that if the entries of $M_n$ are bounded, then the semicircle law holds on intervals of scale $log n/n$. As a consequence, we obtain the delocalization result for the eigenvectors, i.e., the upper bound for the infinity norm of unit eigenvectors corresponding to eigenvalues in the bulk of spectrum, is $O(sqrt{log n/n})$. The bound is the same as the infinity norm of a vector chosen uniformly on the unit sphere in $R^n$. We also study the local version of Marchenko-Pastur law for random covariance matrices and obtain the optimal upper bound for the infinity norm of singular vectors. This is joint work with V. Vu. In the last chapter, we discuss the delocalization properties for the adjacency matrices of ErdH{o}s-R'{e}nyi random graph. This is part of some earlier results joint with L. Tran and V. Vu.
KW - Mathematics
KW - Random matrices
KW - Eigenvectors
LA - eng
ER -