  The paper gives a general condition on permutations, condition under which a
semicircular matrix is free independent, or asymptotically free independent
from the semicircular matrix obtained by permuting its entries. In particular,
it is shown that semicircular matrices are asymptotically free from their
transposes, a result similar to the case of Gaussian random matrices. There is
also an analysis of asymptotic second order relations between semicircular
matrices and their transposes, with results not very similar to the commutative
(i.e. Gaussian random matrices) framework. The paper also presents an
application of the main results to the study of Gaussian random matrices and
furthermore it is shown that the same condition as in the case of semicircular
matrices gives Boolean independence, or asymptotic Boolean independence when
applied to Bernoulli matrices.
