The semicircle law for matrices with dependent entries

Abstract

We investigate the spectral distribution of random matrix ensembles with correlated entries. The matrices considered are symmetric, have real-valued entries and stochastically independent diagonals. Along the diagonals the entries may be correlated. We show that under sufficiently nice moment conditions and sufficiently strong decay of correlations the empirical eigenvalue distribution converges almost surely weakly to the semi-circle law. The present note improves an earlier result (see [Friesen and Löwe, J. Theor. Probab., 2011]) by the authors using similar techniques.