On some Gaussian ensembles of Hermitian matrices

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Abstract
A Gaussian ensemble of Hermitian matrices with an arbitrary ratio of their symmetric and anti-symmetric parts is studied. Similarly a Gaussian ensemble of Hermitian quaternion matrices with an arbitrary ratio of their self-dual and anti-self-dual parts is studied. Analytical expressions for the correlation and cluster functions as well as their limits when the order of the matrices is large, are derived for both ensembles.