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On the lower bound of the spectral norm of symmetric random matrices with independent entries. (English) Zbl 1189.15046

Summary: We show that the spectral radius of an \(N\)-dimensional random symmetric matrix with i.i.d. bounded centered but non-symmetrically distributed entries is bounded from below by \(2\sigma - o(N^{-6/11+\epsilon })\), where \(\sigma ^{2}\) is the variance of the matrix entries and \(\epsilon > 0\) is an arbitrary small positive number. Combining with our previous result from [J. Stat. Phys. 129, No. 5–6, 857–884 (2007; Zbl 1139.82019)], this proves that for any \(\varepsilon > 0\) one has \(\|A_{N}\| = 2\sigma + o(N^{-6/11+\varepsilon})\) with probability going to 1 as \(N\) goes to infinity.

MSC:

15B52 Random matrices (algebraic aspects)
15A18 Eigenvalues, singular values, and eigenvectors
15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
60C05 Combinatorial probability

Citations:

Zbl 1139.82019