### abstract

- The average density of states for a large class of N× N banded and sparse random matrices is shown to obey a semi-circle law. The banded matrices belonging to this class are restricted in several ways: 1) they are both real and symmetric, 2) matrix elements are independent random variables with zero average, 3) the variance of the matrix elements, σ ij 2, decays monotonically away from the diagonal, 4) σ ij depends on| ij| alone and the range over which it significantly varies, δy, satisfies 1 Lt δy Lt N. On the other hand, the sparse matrices for which this results, are obtained by permuting the variance in each of the rows of the banded matrices.