# Local circular law for random matrices

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We consider powers of random matrices with independent entries. Let X, i, j â‰¥ 1, be independent complex random variables with E X = 0 and E| X| = 1, and let X denote an n Ã— n matrix with | X| âˆ’ X for 1 â‰¤ i, j â‰¤ n. Denote by $ s_1^{(m)}\geq \ldots \geq s_n^{(m) } $ the...

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Consider N Ã— N Hermitian or symmetric random matrices H where the distribution of the ( i, j) matrix element is given by a probability measure Î½ with a subexponential decay. Let $${\sigma_{ij}^2}$$ be the variance for the probability measure Î½ with the normalization property that...

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There are two parts in this paper. In the first part we construct the Markov chain in random environment (MCRE), the skew product Markov chain and pâ€“ $$ \ifmmode\expandafter\vec\else\expandafter\vecabove\fi{\theta } $$ chain from a random transition matrix and a twoâ€“dimensional...

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Given a metric space with a Borel probability measure, for each integer N, we obtain a probability distribution on N Ã— N distance matrices by considering the distances between pairs of points in a sample consisting of N points chosen independently from the metric space with respect to the...