Some mathematically simple ensembles of random matrices which represent Hamiltonians with a small time-reversal-noninvariant part

Summary  Some of the statistical properties corresponding to Gaussian ensembles of random matrices whose matrix elements are not all functionally independent are investigated. Three types of ensembles are studied. First, an ensemble whose matrix elements are all real is examined. Next, an ensemble whose matrix elements are not all real, but whose off-diagonal elements have real and imaginary parts which are of the same size on the average is studied. Finally, ensembles with of order-N and-N 2 nonzero imaginary parts of arbitrary size are investigated. Each of the above ensembles is compared with the corresponding Gaussian ensemble in which all of the nonzero matrix elements are functionally independent.